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Doctoral Thesis

Optical Detection of the Hyperfine Interaction in a PositivelyCharged Self-Assembled Quantum Dot

Author(s): Studer, Priska Barbara

Publication Date: 2014

Permanent Link: https://doi.org/10.3929/ethz-a-010209092

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DISS. ETH NO. 21916

Optical Detection of the Hyperfine Interaction

in a Positively Charged

Self-Assembled Quantum Dot

A dissertation submitted to the

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

Priska Barbara STUDER

Dipl. Phys.ETH Zurich

born 28th of October 1984

citizen of Schupfheim LU

Accepted on the recommendation of :

Prof. Dr. Atac Imamoglu, examinerProf. Dr. Klaus Ensslin, co-examiner

2014

c

Abstract

In this dissertation, optical studies of the interaction between either a single hole

spin or a single electron spin confined to a self-assembled InGaAs quantum dot

(QD) with the nuclear spin ensemble of the host material are presented. QDs are

semiconductor structures that allow for a charge carrier confinement in all spatial

directions; this leads to a quantization of the electronic states. Dipole allowed tran-

sitions between the electronic states enable an optical spectrum similar to the one

of atoms. In fact, a QD consists of 105 atoms with nonzero nuclear spin. A single

hole spin or electron spin confined to the QD couples to this ensemble of nuclear

spins via hyperfine interaction. Optical orientation of the electron spin polarization

can be transferred to nuclear spins via hyperfine interaction, leading to dynamic

nuclear spin polarization (DNSP). On the other hand the nuclear spin polarization

acts back on the electron spin, causing an energy splitting of the electronic spin lev-

els. This characteristic can be exploited to optically measure DNSP by performing

spectroscopy.

In the first part of this thesis, we investigate the evolution of nuclear spin polar-

ization in a transverse magnetic field via a single QD electron spin. While in a

magnetic field along the growth direction a pronounced DNSP is expected, in a

transverse magnetic field the nuclear spin polarization should vanish because of the

Larmor precession of the electron. Experimentally, nuclear spin polarization com-

pensates transverse magnetic fields Bx up to ∼1.7T. Therefore, the orientation of

the nuclear spin polarization has to be mainly perpendicular to the optically de-

fined electron spin polarization. To further investigate this anomalous behavior, we

develop a pump probe technique that allows to fully characterize the nuclear spin

polarization as function of Bx by resonant spectroscopy. The aim is to find out

whether the evolution of nuclear spin polarization in transverse magnetic fields can

be mapped to quantum optics models. We find that the nuclear spin polarization

in transverse magnetic fields strongly depends on the sample structure and is very

likely an inherent property of self-assembled InGaAs QDs due to their strain.

In the second part of this thesis, coherence properties of single hole spins confined

in a QD are studied with time resolved measurements. By optically charging the

QD in an n-doped field-effect structure we can benefit from the superior electronic

properties of such devices. We find a coherence time T ∗2 of 240ns which is an or-

d

der of magnitude larger than what was previously measured in p-doped samples.

Additionally, we observe that the hole spin coherence measurements are strongly

influenced by nuclear spins.

e

Zusammenfassung

In dieser Arbeit werden optische Studien der Interaktion zwischen einem einzel-

nen Lochspin oder einem einzelnen Elektronenspin, der in einem selbstorganisier-

ten InGaAs Quantenpunkt (QP) eingeschlossen ist, mit dem Kernspinensemble des

umgebenden Materials vorgestellt. Quantenpunkte sind Halbleiterstrukturen, die ei-

ne dreidimensionale raumliche Eingrenzung von Ladungstragern erlauben, was eine

Quantisierung der Elektronenzustande ermoglicht. Erlaubte Dipolubergange zwi-

schen den Elektronenzustanden fuhren zu einem ahnlichen optischen Spektrum wie

in einem Atom. Im Gegensatz zu einem Atom besteht ein Quantenpunkt jedoch

aus 105 Atomen, die alle einen endlichen Kernspin besitzen. Ein einzelner Lochspin

oder Elektronenspin im Quantenpunkt koppelt zu diesem Kernspinensemble uber

die Hyperfeinwechselwirkung. Deshalb kann die optisch verursachte Polarisation des

Elektronenspins auf die Kernspins ubertragen werden, was zu einer dynamischen

Kernspinpolarisation (DKSP) fuhrt. Andererseits bewirkt die Hyperfeinwechselwir-

kung eine Aufspaltung der Energiezustande des Elektronenspins, wenn das Elektron

in Kontakt mit den spinpolarisierten Kernen steht. Diese Eigenschaft wird verwen-

det, um den Grad der DKSP mit Hilfe von spektroskopischen Methoden optisch zu

bestimmen.

Im ersten Teil dieser Arbeit wird die Entwicklung von Kernspinpolarisation in einem

transversalen Magnetfeld mit Hilfe von einem einzelnen Elektronenspin im Quan-

tenpunkt untersucht. Wahrend in einem Magnetfeld entlang der Wachstumsrich-

tung des QPs eine starke DKSP erwartet wird, sollte die Kernspinpolarisation in

einem transversalen Magnetfeld aufgrund der Larmorprazession des Elektrons ver-

schwinden. Experimentell kann eine Kompensation des externen Magnetfeldes Bx

durch die Kernspinpolarisation bis zu ∼1.7T festgestellt werden. Dazu muss die

Kernspinpolarisation mehrheitlich senkrecht zur optisch definierten Polarisation der

Elektronenspins orientiert sein. Um diesen ungewohnlichen Polarisationsmechanis-

mus der Kernspins zu studieren, wird eine Pump-Probe Technik entwickelt, die eine

vollstandige Charakterisierung von der Kernspinpolarisation als Funktion von Bx

mit Hilfe von resonanter Spektroskopie erlaubt. Das Ziel dabei ist herauszufinden,

ob die Entwicklung der Kernspinpolarisation in einem transversalen Magnetfeld mit

quantenoptischen Modellen beschrieben werden kann. Wir stellen fest, dass die Kern-

spinpolarisation in transversalen Magnetfeldern sehr stark von der Probenstruktur

f

abhangt und sehr wahrscheinlich eine inharente Eigenschaft von selbstorganisierten

InGaAs QP aufgrund ihrer Gitterverspannung ist.

Im zweiten Teil dieser Dissertation werden die Koharenzeigenschaften von einzel-

nen Lochspins in QP mit zeitaufgelosten Messungen untersucht. Indem der QP

optisch mit einem einzelnen Lochspin in einer n-dotierten Feldeffektstruktur ge-

laden wird, konnen wir von den ausgezeichneten elektronischen Eigenschaften dieser

Struktur profitieren. Wir messen eine Koharenzzeit T ∗2 von 240ns, die verglichen

mit vorhergehenden Messungen in p-dotierten Strukturen um eine Grossenordnung

langer ist. Daruber hinaus stellen wir eine starke Beeinflussung der Lochspin

Koharenzmessungen durch Kernspins fest.

Contents g

Contents

Title a

Abstract d

Zusammenfassung e

Contents g

List of symbols and abbreviations i

1. Introduction 1

1.1. Motivation and scope of this thesis . . . . . . . . . . . . . . . . . . . 2

2. Self-assembled Quantum Dots and Nuclear Spins 5

2.1. Charge tunable self-assembled Quantum Dots . . . . . . . . . . . . . 5

2.1.1. Growth of self-assembled quantum dots . . . . . . . . . . . . . 6

2.1.2. Quantum dot level scheme and optical selection rules . . . . . 7

2.1.3. Charge control . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4. Quantum confined Stark effect . . . . . . . . . . . . . . . . . . 11

2.2. Nuclear Spins in Quantum Dots . . . . . . . . . . . . . . . . . . . . . 13

2.2.1. Electron and Hole Spin System . . . . . . . . . . . . . . . . . 13

2.2.2. Nuclear Spin System . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3. Isotropic and Anisotropic Hyperfine Interaction . . . . . . . . 16

2.2.4. Electron Spin Decoherence . . . . . . . . . . . . . . . . . . . . 18

3. Experimental Methods 21

3.1. Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2. Resonant Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . 22

3.3. Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4. Pump Probe Measuring Methods . . . . . . . . . . . . . . . . . . . . 24

3.4.1. Optical detection of Nuclear Spin Polarization . . . . . . . . . 25

3.4.2. Pump Probe Method with Resonant Rayleigh Scattering . . . 26

3.4.3. Pump probe Method with Resonance Fluorescence . . . . . . 34

h Contents

4. Nuclear Spin Polarization in Transverse Magnetic Fields 37

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2. Hanle experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3. Anomalous Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4. Hanle measurements for samples with different electron spin lifetimes 41

4.5. Resonant measurements on a 35nm tunnel barrier sample . . . . . . . 45

4.6. Resonant measurements on a 25nm tunnel barrier sample . . . . . . . 46

4.6.1. Gate properties . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6.2. Resonance fluorescence background . . . . . . . . . . . . . . . 48

4.6.3. Extraction of the resonance positions . . . . . . . . . . . . . . 49

4.6.4. Transition between Overhauser shift and hole Zeeman splitting 50

4.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5. Coherence of Optically Generated Hole Spins 59

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2. Spectroscopy on X+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3. Initialization and Readout . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4. Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5. Ramsey measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6. Spin Echo measurements . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A. Appendix A

A.1. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . A

B. Bibliography I

Acknowledgements XI

Curriculum Vitae XIII

List of Figures XV

List of symbols and abbreviations i

List of symbols and abbreviations

gel,h,ex g-factor (for electron, hole and exciton, respectively)

j Total angular momentum (spin and orbital)

l Orbital angular momentum

Ai Hyperfine coupling constant of nucleus i

Bel Knight field

Bext External magnetic field

Bloc Local dipolar nuclear field

Bnuc Overhauser field

EF Fermi energy

Ii, I i Nuclear spin operator and quantum number

N Number of quantum dot nuclei

Q Nuclear quadrupolar moment

S, S Electron spin operator and projection on z−axis

T1 Spin relaxation time

T2,(T ∗2 ) Spin dephasing time (ensemble averaged)

Vg Applied gate voltage

γi Gyromagnetic ratio of nucleus i

ν0 Unit cell volume

ρ+c (ρ−c ) Circular polarization of PL light (under σ+- (σ−-) polarized excitation

σ+(σ−) Circular light polarization of positive (negative) helicity

σx Linear light polarization along x

ωQ Quadrupolar coupling strength

Ψ Electron wave function

Ωel Electron Larmor frequency

∆EOS Overhauser shift

∆EZel,e,ex Zeeman splitting (for electron, hole and exciton, respectively)

↑, ↓ (⇑, ⇓) Electron (hole) spin up, electron (hole) spin down

j List of symbols and abbreviations

1eV = 1.6 · 10−19 J 1 electronvolt

e = 1.6 · 10−19 A·s Electron charge

g0 = 2 Free electron g-factor

~ = 6.582 · 10−10 µeVs Reduced Planck’s constant

µB = 58 µeV/T Bohr magneton

k = 86 µeV/K Boltzmann constant

µ0 = 4π · 10−7N/A2 Permeability of free space

As Arsenic

CB Conduction band

CCD Charge coupled device (camera)

DNSP Dynamical nuclear spin polarization

CPT Coherent population trapping

FWHM Full width at half maximum

Ga Gallium

HH Heavy hole

In Indium

LH Light hole

MBE Molecular beam expitaxy

PL Photoluminescence

PLE Photoluminescence excitation

QD Quantum dot

QI Quadrupolar interaction

RF Resonance fluorescence

SIL Solid immersion lens

SNR Signal to noise ratio

Ti Titanium

VB Valence band

Xn Exciton of charge n

Introduction 1

1. Introduction

We all use modern electronic devices in daily life for work, communication and

entertainment. Year by year, the computing power of such devices grows. One

of the main reasons for this development is the miniaturization of the electronic

components. Hence an increasing amount of functionality can be realized on the

same area of the computer chip. As a consequence a computer processor consists of

millions of transistors, where the smallest have a gate length of 10nm. The ability

to fabricate devices at such small length scales has also inspired many research

areas. The growth and study of objects where at least one of the spatial dimensions

extends to only a few nanometers is called Nanotechnology. If objects are made

smaller and smaller, the classical description fails at low temperature and quantum

mechanical effects play an important role.

The study of charge carriers that are confined in low dimensions led to the discovery

of interesting electrical and optical properties such as the Aharanov-Bohm effect [1],

Coulomb blockade [2] and the quantum Hall effect [3]. In this thesis we focus on

semiconductor structures with a confinement in all spatial directions, quantum dots

(QD). On one hand, a QD shows a discrete energy spectrum and optical properties

like an atom, on the other hand it is an object consisting of 104 − 105 atoms. The

confinement allows for a study of the interaction of a single electron or hole with

its environment. One example is the interaction with a nearby electron or hole

reservoir that results in interesting many-body physics [4–6]. Furthermore, the

electron spin in a QD couples to the nuclear spins of the atoms the QD consists of

via hyperfine interactions [5, 7, 8].

Single electrons confined in a QD are considered as promising candidates for

the use as spin qubits in quantum information processing [9, 10]. Compared

to atoms, the on-chip realization is promising for the scalability of the system.

Additionally, QDs exhibit long spin lifetimes and coherence times due to the weak

interaction with the environment [11]. This is absolutely crucial for quantum

information storage. Last but not least, the coherent optical control of electron

spins in self-assembled QDs allows for a fast manipulation of the quantum state on

picosecond timescales [12, 13]. Therefore a prospering research field was established

which addresses the problem of initialization, manipulation and readout of the

2 Introduction

electron spin [14–16].

However, a single electron spin in a QD is not completely isolated from its envi-

ronment. On the one hand, coupling to the environment is needed to manipulate

and read out its state and on the other hand this coupling constitutes a source of

decoherence. A particular interaction is the strong coupling of the electron spin to

nuclear spins. Since the nuclear spins are randomly oriented in a QD the electron

spin experiences a fluctuating effective magnetic field Beff. As a consequence,

nuclear spins are the dominant source of decoherence for electron spins [17]. It

is well known that a long coherence time is important for quantum information

processing, therefore the strong coupling to nuclear spins might be a drawback.

One possible way to prolong the coherence time is preparing nuclear spins in a state

which exhibits less fluctuations [18]. Additionally, by controlling nuclear spins it

should be possible to influence the electron spin decoherence. For the realization

of such a scenario, the interaction between electron and nuclear spins has to be

studied in detail.

The hole on the other hand, which is a missing electron form the valence band,

exhibits very little interaction with the nuclear spins. Because of the p-symmetry

of the hole wave function the effective overlap of holes and the nuclei is strongly

reduced compared to electrons. Hence, the hyperfine interaction to nuclear spins is

weak, leading to longer coherence times [19].

1.1. Motivation and scope of this thesis

This thesis focuses on the optical study of nuclear spin interactions either with

a single electron or a hole that is confined in a self-assembled InGaAs QD and is

structured as follows.

Chapter 2 is an introduction to self-assembled InGaAs QDs and nuclear spins in

QDs. In a first part we describe the growth of self-assembled QDs and the result-

ing discrete energy levels with their optical selection rules. Furthermore the charge

control is discussed, which is realized by embedding the QDs in a diode structure.

For the study of nuclear spins it is important to know the interactions in the nuclear

spin system as well as the coupling to a single electron or hole spin. We discuss the

corresponding interactions in the second part of chapter 2. Finally, the influence of

the electron - nuclear spin coupling on coherence properties of the electron spin are

lined out.

To gain insights into the interaction between nuclear spins and the spin of a sin-

gle charge carrier confined in a QD, the spin properties of the single charge carrier

are studied optically. A variety of different spectroscopy methods are introduced in

chapter 3. While non-resonant excitation methods are important for a first charac-

1.1. Motivation and scope of this thesis 3

terization of the QD, resonant spectroscopy techniques, namely resonant Rayleigh

scattering and resonant fluorescence, allow for a direct access to optical transitions

with high spectral resolution. Polarization of nuclear spins leads via the hyperfine

interaction to a shift of the electron energy levels, the so-called Overhauser shift.

The second part of chapter 3 covers the development of a pump probe method that

allows for a resonant measurement of the Overhauser shift. Especially the increased

spectral resolution compared to non-resonant measurement methods is promising to

get new insights in the interactions between the electron spin and the nuclear spin

reservoir.

Chapter 4 focuses on the evolution of nuclear spin polarization in a transverse mag-

netic field Bx. We will show that the nuclear spin polarization is very likely to

cause a cancellation of Bx for the electron spin up to Bcrit = 1.7T. This is rather

unexpected, since nuclear spins are optically polarized along the z-axis resulting in

an effective magnetic field Beffz ∼ 500mT that is much smaller than the measured

Bcrit = 1.7T. There are several indications that the generated nuclear spin polariza-

tion is rotated to the x-axis [20]. To further investigate the nuclear spin evolution

during this process, we measured the Overhauser shift resonantly for two different

sample structures.

In chapter 5, time resolved measurements on the coherence properties of a single hole

spin confined in a QD are presented. In contrast to electrons, holes interact weakly

with nuclear spins, which leads to longer coherence times. To additionally prolong

the coherence time, we created the hole optically in an n-doped sample, which re-

sults in less charge noise compared to p-doped samples. We find a coherence time

of T ∗2 = 240ns. Spin echo measurements reveal a T2 of 1µs.

Self-assembled Quantum Dots and Nuclear Spins 5

2. Self-assembled Quantum Dots

and Nuclear Spins

In the first part of this chapter the basic principles of self-assembled In-

GaAs QDs are explained. We start with a section about the growth of self-

assembled QDs. Subsequently, the consequences of such a quantum confine-

ment potential in three spatial directions and the optical selection rules are

discussed. Furthermore the diode-structure used for controlling the number

of excess charges in the QD is presented. In the second part of this chapter,

the theoretical background of the nuclear spin system and a single electron

or hole spin confined in a QD are introduced. While the coupling of the

electron-nuclear spin system is dominated by the contact hyperfine interac-

tion, the hole couples through the anisotropic dipolar hyperfine interaction to

the nuclear spin ensemble. Finally, we will discuss the influence of nuclear

spins on the electron spin decoherence.

2.1. Charge tunable self-assembled Quantum Dots

A quantum dot (QD) is an artificial structure that exhibits a strong confinement

in all three spatial dimensions. In other words, it is a zero dimensional object con-

fining single charge carriers. There are different ways to realize such a confinement.

One possibility is to deplete a two-dimensional electron gas with electric gates or by

oxidation, which results in a spatial confinement for electrons. Such a QD is usually

coupled via tunnel barriers to electronic reservoirs which allow for a deterministic

charging with single electrons. To investigate the physical properties, current and

voltage probes are attached to these reservoirs. Therefore electronic properties can

be measured [21].

A second possibility is to generate confinement by a semiconductor surface which is

realized in semiconductor nanocrystals [22]. These QDs are grown by crystallization

in a colloidal solution, and are therefore called colloidal QDs [23]. Nanocrystals are

optically active, the transition energy depends on the size of the crystal, which is

a growth parameter. On the one hand, the performance of colloidal quantum dots

6 Self-assembled Quantum Dots and Nuclear Spins

for quantum optics experiments is strongly affected by spectral diffusion leading to

a broadening of the optical linewidth of several 100µeV [24]. Additionally, colloidal

QDs are suffering from blinking and bleaching. On the other hand, the chemical

stability allows to use them as fluorescence markers in biology [23].

In self assembled semiconductor QDs [25] the charge confinement is given by a

combination of semiconductor materials with different band gaps and a type I in-

terface (see section 2.1.1). Compared to colloidal QDs, the optical line widths are

much narrower and near life-time limited (∼ 1.6µeV) [26]. Additionally, a single

self-assembled QD behaves like a quantum emitter which manifests in perfect anti-

bunching of the emitted photons [27]. Hence a denotation as artificial atom is very

common. However, a self-assembled QD is still a mesoscopic object which consists

of 104 − 105 atoms. Many interesting physical effects arise due to the interaction of

the confined electron with the host material of the QD and the resulting strain from

the growth process.

2.1.1. Growth of self-assembled quantum dots

The InGaAs QDs studied in this work were grown by molecular beam epitaxy

(MBE). This method allows depositing single crystals with a rate of less than

3000nm per hour and therefore a very precise control over the material composi-

tion is achieved. MBE takes place in high vacuum and the deposited particles are

heated in separate effusion cells. The gaseous particle beams condense subsequently

on the wafer. For growth of InGaAs QDs, a GaAs wafer is heated to 600C under

ultra-high vacuum. First, a buffer layer of several 100nm of GaAs is grown to clean

the surface of the wafer. In a second step, the evaporated gallium (Ga) particle

beam is replaced with indium (In). There is a large lattice mismatch between GaAs

and InAs of 7%, which leads to an abrupt change of the growth characteristics at

a critical layer thickness of InAs. After an initially two dimensional growth of InAs

(Fig. 2.1 (a)) the mechanical strain is released at 1.7 monolayers by the formation

of three dimensional InAs islands (Fig. 2.1 (b)). At this critical point it is energeti-

cally favorable to minimize strain energy by increasing surface energy. The growth

process is stopped after 2.7 monolayers of InAs. A continuation would again result

in a two dimensional layer of InAs with crystal defects which relax the strain. The

formed InAs QDs are randomly distributed and have a diameter of 20-30nm and a

height of 10nm. This growth process, which results in the spontaneous formation of

InAs QDs, is called Stranski-Krastanow method [28].

Typical transition wavelengths for these QDs are at 1100nm, which is a rather

inconvenient spectral range for efficient silicon based photodetectors. Therefore the

QDs are partially covered with GaAs (Fig. 2.1 (c)) and the temperature is increased.

In this annealing process the intermixing between Ga and InAs is increased, result-

2.1. Charge tunable self-assembled Quantum Dots 7

(a) (b) (c) (d) (e)

GaAs

InAs

Figure 2.1.: Growth of self-assembled QDs. (a) Two dimensional growth ofInAs. (b) Strain driven formation of InAs islands after 1.7 monolayers. (c) Partialcapping layer of GaAs is grown. (d) Annealing leads to a reduced height of theQDs. (e) Protective overgrowth with GaAs.

ing in a smaller QD height (Fig. 2.1 (d)). The stronger confinement in the growth

direction leads to a shift of the transition wavelength to the blue i.e. 950nm, which

is a much better spectral range for silicon based photodetectors. This method to

shift the wavelength is known as the partially covered island (PCI) technique [29].

As a last step the InAs QD are covered with a GaAs layer to avoid surface effects

(Fig. 2.1 (e)).

Even after the formation of InAs droplets, there is a thin two dimensional layer of

the same material remaining, which is called the wetting layer. This quantum well

structure can be optically accessed and is energetically shifted compared to the QD

spectra. The emission of the wetting layer is expected at 840nm-860nm.

Self-assembled QDs exhibit a distribution in size, shape, In and Ga content. This

leads to a distribution of their transition energies. Usually, an ensemble of QD shows

an inhomogeneous broadening in the order of 10meV [30], compared to the natural

line with of ∼ 1µeV of a single QD [31]. To resolve the line width isolation of a

single QD is desirable. Additionally, the growth process allows to control the QD

density as follows. To assure a homogeneous result of the growth process, the wafer

is rotated. During the formation of the QDs the rotation of the wafer is stopped.

Therefore a gradient of the QD density develops according to the distance from the

indium source. By choosing the corresponding piece of the wafer, a low density

region can be selected to perform studies on single QDs.

2.1.2. Quantum dot level scheme and optical selection rules

The surrounding of the InAs QD with GaAs that exhibits a larger band gap leads

to a quantum confinement potential in all three dimensions. In Fig. 2.2 the potential

landscape perpendicular to the growth direction of the QD is shown, which can be

approximated by a 2D harmonic potential. Several discrete energy states that can


be occupied by electrons in the conduction band (CB) and holes in the valence band

(VB) are formed. In analogy to atomic states, the discrete energy states of the

confining potential are labeled s, p, d,... Furthermore, the quantum confinement

leads to a separation of the hole band into a heavy hole (HH) band and a light hole

(LH) band [32] that is additionally enhanced by the strain in self-assembled QDs.

The LH band is shifted by more than 10meV to lower energies such that only the

HH band is considered for further studies on QDs.

The Bloch wave function of the lowest energy states in the conduction band exhibit

an s-type character, which means that the total angular momentum of an electron

occupying the CB is provided by its spin Sz = ±1/2. On the other hand the highest

energy states of the valence band have a p-type wave function, leading to an angular

momentum of 1. Therefore the total angular momentum projection of the HH spin

is Jz = ±3/2. Further consequences of the different wave function symmetries of

electrons and holes will be discussed in detail in section 2.2.

An electron can be lifted from the valence band to the conduction band in the QD

r (nm)

-100 -50 0 50 100

~ 300 meV

~ 150 meV

1445 meVs

p

d

spd

CB

VB

Harmonic

approximation:

Lateral confinement:

s

s p

p

VB

CB

e

h

electron

hole 100 -50 -100 0 50

r (nm)

1519

(a) (b)

Figure 2.2.: Confinement in an InGaAs QD. (a) Confinement potential of anInGaAs QD perpendicular to the growth plane. Discrete energy states are labeledwith s,p,d,... similar to atomic states. (b) An approximation of the confinementpotential is realized with a 2D harmonic oscillator where the energy states are sepa-rated by ~ωe for electrons in the conduction band (CB) and by ~ωh for holes in thevalence band (VB). Figure taken from [33].

i.e. by absorbing a photon. The electron in the conduction band is bound to the

hole in the valence band by the Coulomb attraction, resulting in a quasi-particle

called exciton. Subsequently a recombination of this electron hole pair can take

place, by emission of a photon. Until now we only considered neutral excitons (X0).

By confining extra charge carriers into the QD other exciton types can be formed. A

negatively charged exciton (X−) is generated if an excess electron is inside the QD.

Typically the emission wavelength for X− is red shifted by ∼6meV as compared to

X0. Similarly a positively charged exciton (X+) can be created with an additional


hole inside the QD. In this case the electron hole recombination energy depends

strongly on the QD and can be a few meV larger or smaller than that of the neutral

exciton. For all following experiments we focused on a positively charged QD.

Based on the twofold spin degeneracy of the electron and the hole inside the QD

four different excitons can be generated for X+. Their total angular momentum

projection is given by Mz = Sz+Jz leading to Mz = ±1 or Mz = ±2. Because of the

conservation of angular momentum, the emitted photon upon exciton recombination

carries the spin of the electron hole pair. Therefore only the exciton with Mz =

±1 can decay radiatively, resulting in a σ+, respectively a σ− polarized photon.

These excitons are called bright. The so-called dark excitons with Mz = ±2 cannot

recombine optically.

For a deterministic charging with a single electron or hole, the QD is exposed to

an electric field and coupled to a nearby electron reservoir. This can be realized by

embedding the QD layer into a field effect structure [34, 35], which is explained in

detail in the following section.

2.1.3. Charge control

The samples studied in this thesis consist of a quantum dot layer which is

embedded in a field effect structure as shown in Fig. 2.3. Similar to QD fabrication,

this electronic structure is grown by MBE. In order to apply an electric field to the

QD, the QD layer is sandwiched between a negatively doped GaAs back contact

and a metallic top gate. A layer of intrinsic GaAs acts as a tunnel barrier between

the back contact and the QD layer. Additionally a blocking barrier, consisting

of AlGaAs on top of the capping layer, is needed to prevent current flow in the

sample. Finally the electronic structure ends with a layer of intrinsic GaAs that

prevents the aluminum in the blocking barrier from oxidation. After the growth

process a semitransparent metallic top gate, consisting of 2nm titanium and 8nm

gold, is deposited on top of the sample.

The electrostatic energy of an electron confined in a QD is determined not only

by the applied voltage Vg between the back contact and the top gate but also by

an offset voltage V 0g , arising from the Schottky interface at the top gate. Since the

Fermi energy EF is pinned to the conduction band (CB) at the back contact, the

applied gate voltage changes the energy of the QD states depending on the lever

arm η = d/b, where d is the distance of the top gate from the back contact and b is

the thickness of the tunneling barrier.

By choosing the gate voltage such that the electronic state of the QD is shifted

below EF an electron tunnels into the QD from the back contact, which acts as

an electron reservoir [35]. Such a deterministic charging of the QD with single

electrons is only possible due to the Coulomb repulsion which prevents a second


GaAs:n+ GaAs:i GaAsAlAs/GaAssuperlattice

quantum dotlayer

bandgap

CB

VB

Ef

back gate

top gate

e(Vg-Vg0)

Figure 2.3.: Field effect structure. Conduction band (CB) and valence band(VB) edge of the field effect structure. The n-doped back gate fixes the Fermienergy EF to the CB. With an externally applied gate voltage (Vg) between the topgate and the back contact, the slope of the band edges can be controlled. Figuretaken from [33].

electron from tunneling into the QD. By applying a more positive Vg the Coulomb

repulsion can be overcome, resulting in a QD charged with two electrons in a singlet

state. Similarly, the p-shell states of the QD can be successively filled.

Until now, we have only considered charging of the QD with electrons. There are

two possibilities to achieve this for holes. Either the back contact is p-doped and

acts as a hole reservoir or the holes are optically generated in an n-doped sample.

For the experiments in this thesis we used the latter. A detailed comparison of the

two methods is given in chapter 5.

In order to charge the QD optically with a single hole, an electron is excited to

the conduction band leaving a single hole in the valence band. Subsequently, both

charge carriers relax into the QD. If the applied gate voltage is chosen such that

the electronic state lies above the Fermi energy, the electron tunnels out of the QD

(Fig. 2.4(b)). The criteria for the hole to stay in the QD not given by EF but rather

by the position of the two dimensional continuum of hole states at the interface to

the superlattice [4, 36]. If the lowest subband energy of the hole continuum is shifted

below the QD hole state, the hole cannot tunnel out of the QD. Consequently the

hole is trapped inside the QD and a positively charged exciton X+ can be created

by adding an optically generated electron hole pair (Fig. 2.4(c)) [37]. This state is

only stable due to the additional binding energy of the second hole that results in

an electronic state below the Fermi energy (Fig. 2.4(d)).

In order to charge the QD with two excess holes, a more negative gate voltage

Vg has to be applied. In this case, also the second electron tunnels out of the

QD because it overcomes the attractive Coulomb energy. Therefore two holes are

trapped inside the QD, which leads to a X2+ state upon optical excitation. The


CB

VB

Ef

(a) (b)

(c) (d)

Figure 2.4.: Charging n-doped samples with a single hole. (a) After theoptical excitation of the electron to the CB, the electron hole pair relaxes into theQD. (b) Since the electronic state of the QD lies above the Fermi energy, the electrontunnels out of the QD. On the other hand the hole remains trapped because of theQD hole state laying below the two-dimensional continuum of states. (c) A secondoptical excitation leads to the formation of a positively charged exciton. (d) ThisX+ state is stable due to the Coulomb energy of the second hole which results inan electronic state laying below the Fermi energy. Figure taken from [33].

number of observed charge states depends strongly on the confinement potential.

While for InAs QD studied by Ediger et al. [38] states from X6+ to X7− were

observed, in Fig. 2.5 charged excitons from X+ to X−3 are shown.

2.1.4. Quantum confined Stark effect

Besides the discrete jumps in the photoluminescence (PL) measurements as a

function of the gate voltage due to the different charging states, a slight shift of the

energy plateaus with the gate voltage is observable in Fig. 2.5. This is due to the

quantum confined Stark effect which arises on the one hand from the permanent

dipole moment dexc of the exciton and on the other hand from the induced dipole

moment. To quantify the latter we introduce the polarizability α. As a consequence


-0.6 -0.4 -0.2 0.0 0.2

974

973

972

971

970

969

X3-X

2-X-

X0

Gate Voltage (V)

Wavele

ngth

(nm

)

X+

Figure 2.5.: Charging diagram measured in PL. Colour plot of the PL intensityas a function of gate voltage and emitted wavelength, red means high and blue lowintensity. The measurement was taken on a 25nm tunneling barrier sample. Thejumps in energy between the different charging states are observable. The differentlycharged excitons are labeled with X+ for a positively charged exciton, X0 for theneutral exciton, X− for the negatively charged exciton and so on.

the exciton energy changes as a function of the effective electric field. This can be

described with the following formula [39]

E = E0 − dexcF − αF, (2.1)

where E0 is the energy of the unperturbed exciton and F is an external electric field.

Especially from the experimental point of view the quantum confined Stark shift is

very beneficial. It allows to scan across the exciton resonance by varying Vg without

changing the laser wavelength.

2.2. Nuclear Spins in Quantum Dots 13

2.2. Nuclear Spins in Quantum Dots

As previously mentioned, a self-assembled QD consists of 104 − 105 atoms and a

single confined charge carrier is not completely decoupled from its environment. In

particular, the confined electron spin in a QD couples to the nuclei of the atoms

that the dot consists of. This leads to several interesting physical phenomena,

for example locking of a QD transition to a resonant pump laser [40, 41] or

bistabilities of the nuclear spin system that exhibit memory effects (non-Markovian

behavior) [42, 43]. Furthermore, the random fluctuations of the nuclear spin system

are the main source of decoherence for a single electron spin confined in a QD [17].

The prospect of prolonging the coherence time to the extent that a single spin

manipulation is possible, was one of the main motivations that led to increased

interest in nuclear spin physics in QDs [18]. Here, we will give an overview of

the electron/hole spin system, the nuclear spin system and the coupling between

the two. Last but not least we consider the influence of these couplings on the

coherence properties of a single electron spin in a QD.

In an externally applied magnetic field, the total Hamiltonian for a single

electron/hole interacting with an ensemble of nuclear spins is given by

Htot = Helz +Hnuc

z +Hihf +Hahf +Hdip +HQ, (2.2)

where Helz and Hnuc

z are the electron and nuclear Zeeman Hamiltonians, respectively.

While for the electron the isotropic hyperfine interaction Hihf is the dominant cou-

pling to the nuclear spin system, the anisotropic hyperfine interaction Hahf describes

the most important interaction between the hole spin and the nuclear spins. Hdip

denotes the nuclear dipole-dipole interaction while HQ are the nuclear quadrupolar

interactions. In the following section we will discuss the different terms in detail.

2.2.1. Electron and Hole Spin System

In the framework of this thesis we studied manly positively charged QDs. While

in chapter 4 the electron spin in the positively charged exciton is investigated, the

experiments in chapter 5 focus on the hole spin in the ground state. Therefore

we will consider the interaction of an electron spin and a hole spin to an external

magnetic field in this section.

For a QD electron spin, the application of a magnetic field ~B leads to a Zeeman

splitting:

Helz = gelµB ~Sel · ~B, (2.3)


where gel is the effective electron g-factor varying from dot to dot between 0.4 to

0.6, µB denotes the Bohr magneton and ~Sel the electron spin operator. In analogy

the Zeeman interaction for the hole is given by

Hhz = −ghµB ~Sh · ~B, (2.4)

with the effective hole g-factor gh and the hole spin operator ~Sh. In contrast to gel,

the effective hole g-factor depends strongly on the magnetic field direction. While

for magnetic fields along the growth direction gh is around 1-1.5, for transverse

magnetic fields gh is between 0 and 0.4 [44, 45].

2.2.2. Nuclear Spin System

The material composition of the considered QDs is mainly given by InAs and has

some additional Ga contributions due to diffusion processes [46]. The nuclear spin

system is therefore given by the naturally occurring isotopes of the three atomic

species: 115In (95.3%), 113In (4.7%), 75As, 69Ga (60.1%), 71Ga (39.9%). The natural

abundance is denoted in brackets. While Ga and As have a total nuclear spin of

3/2, In has a spin of 9/2. In a static magnetic field, each nucleus i experiences the

following Zeeman interaction:

Hnucz = −

∑i

γi~Ii · ~B, (2.5)

with γi the gyromagnetic ratio and ~I i the spin operator of the ith nucleus.

Nuclear Dipolar Coupling

The nuclear dipole-dipole interaction is an important coupling within the nuclear

spin system. A nuclear magnetic moment γiIi will produce a dipolar magnetic field

which acts on a nuclear spin j. This interaction is strongly dependent on the relative

position rij between two nuclear spins i and j [47].

H i,jdip =

µ0γiγj4πr3

ij

(I i · Ij − 3

(I irij)(Ijrij)

r2ij

)(2.6)

The fluctuating local magnetic field created at the nucleus j by the other nuclei can

be estimated from the bulk value in GaAs, Bloc ≈ 1.5 · 10−4T [48].

The nuclear dipolar coupling (2.6) can be decomposed into ”secular” parts which

commute with Hnucz (energy conserving) and into ”non-secular” parts.


The ”secular” parts have the following form

Hsecdip ∼ I izI

jz −

1

4

(I i+I

j− + I i−I

j+

). (2.7)

Clearly, Hsecdip conserves the nuclear spin. The second term in (2.7) describes flip-flops

between nuclear spins of the same species at different sites, which leads to nuclear

spin diffusion.

”Non-secular” terms are only important for Bext 5 Bloc. They do not conserve

the total spin of the nuclei, it is transferred to the crystal lattice. Therefore a

depolarization within tnuc ≈ 10− 100µs takes place [21], if quadrupolar interactions

are absent (see following section).

Nuclear Quadrupolar Interactions

After looking into magnetic interactions between nuclear spins we will now focus

on the influence of an electric field on nuclei. The electrostatic energy of a localized

charge distribution ρ(~x) placed in an external potential φ(~x) can be written with a

multipole expansion [49]:

W = qφ(0)− pE(0)− 1

6

∑i

∑j

QijδEjδxi

(0) + ... (2.8)

p =

∫x′ρ(x′)d3x′ dipole moment

Qij =

∫(3x′ix

′j − r′2δij)ρ(x′)d3x′ quadrupole moment

Due to symmetry reasons, the nuclei do not have an electric dipole moment and are

therefore insensitive to homogeneous fields [47]. Nuclear spins with I > 1/2 pos-

sess a finite electric quadrupole moment which couples to an electric field gradient.

Generally the quadrupole moment describes the ellipticity of a charge distribution

in a nuclei and is strongly dependent on the material.

If the nuclear environment has cubic symmetry (as in bulk GaAs), there is no elec-

tric field gradient and therefore no quadrupolar coupling [47, 50]. Crystal strain in

self-assembled QDs gives rise to nonzero electric field gradients at the position of

the nuclei. Assuming the electric field gradients have cylindrical symmetry and are

oriented along z′, the quadrupolar interactions can be described by

HQ =~ωQ

2

(I2z′ −

I(I + 1)

3

)(2.9)

where ωQ is the quadrupolar coupling strength and Iz′ the angular momentum

projection on the z′-axis [47]. Without any external field, the (2I + 1) spin lev-


els are paired into mz′ = ±1/2,±3/2... ± I and separated from each other by

1, 2, ...(I − 1/2)~ωQ.

By setting gNµNBQ = ~ωQ, where gNµN =∑

i γi, an effective quadrupolar magnetic

field can be calculated. In self-assembled InAs QDs BQ is estimated and measured

to be several 100mT [51–53]. This nonlinear energy splitting between different spin

levels leads to a strong suppression of nuclear spin diffusion and therefore to long

nuclear polarization storage times [52].

2.2.3. Isotropic and Anisotropic Hyperfine Interaction

Isotropic Hyperfine Interaction

The dominant interaction between a single QD electron spin and an ensemble of

nuclear spins is the isotropic hyperfine interaction [17, 54]. The Hamiltonian is given

by

Hihf = ν0

∑k

Ak|ψ(~Rk)|2~S · ~Ik, (2.10)

where ν0 denominates the atomic volume of InAs , Rk denotes the position of the

kth nucleus and ψ(Rk) is the electron envelope wave function. The hyperfine cou-

pling constant is Ak = 16π3µ0g0µB~γk|u(~Rk)|2 with u(~Rk) the Bloch amplitude at the

position of the nuclei and g0 the free electron g-factor. There are only non-vanishing

terms of Ak if the Blochfunction is non zero at the position ~Rk of the nuclei. This

focuses the attention on the wave function in a QD. In GaAs the top of the valence

band has a p-like symmetry and the bottom of the conduction band has an s-like

symmetry. Contrary to p-like symmetry, the probability of an s-like wave function

symmetry does not disappear at the site of the nuclei. Therefore the isotropic hyper-

fine interaction has only non-vanishing contributions for electrons in the conduction

band. The strength of the total hyperfine constant of the main components indium,

gallium and arsenide are AIn ∼= 56µeV, AGa ∼= 42µeV and AAs ∼= 46µeV [55].

The Hamiltonian of the isotropic hyperfine interaction (2.10) can be divided into a

static and a dynamical part [56]:

=⇒ ~S · ~Ik = IkzSz︸︷︷︸static part

+1

2

(Ik+S− + Ik−S+

)︸︷︷︸dynamical part

. (2.11)

S+ = Sx + iSy raising operator

S− = Sx − iSy lowering operator(2.12)

The static part of the effective Hamiltonian affects energies of the electron and

nuclear spins leading to an ”effective magnetic field”, which is only experienced by


the electron due to spin polarized nuclei (Overhauser field) or by the nuclei due to

a spin polarized electron (Knight field).

– Overhauser Field

Hstaticihf =

(∑k

AkIkz

)Sz =⇒ Bnuc =

1

g∗elµB

∑k

AkIkz (2.13)

A fully polarized nuclear spin system in GaAs can lead to an effective magnetic

field of several Tesla [57], which is experienced by the electron.

– Knight Field

Hstaticihf =

∑k

(AkS

kz

)Ikz =⇒ Bk

el = − 1

gNµNAkSz (2.14)

The maximal effective magnetic field due to a polarized electron spin is

Bel,max = 13mT for Ga and Bel,max = 22mT for As [57].

The dynamical part of the effective Hamiltonian allows transferring angular momen-

tum between the two spin systems. If the external applied magnetic field Bext is

larger than the local magnetic field Bloc, direct electron nuclear flip-flop processes

will vanish due to the large energy mismatch of the electronic and nuclear Zeeman

splitting (electron: g∗elµB = 10−2/T meV, nuclei: gNµN = 10−5/T meV). In this case

higher order nuclear spin flips can still take place via an excited or virtual state [58].

Anisotropic Hyperfine Interaction

In addition to the contact hyperfine interaction discussed in the previous section,

the electron or hole can couple to a dipole-like hyperfine interaction to the nuclear

spins. The Hamiltonian of this anisotropic hyperfine interaction is given by

Hahf =µ0

4πg0µB

∑k

γk3(~nk · ~S)(~nk · ~Ik)− ~S · ~Ik

R3k

, (2.15)

where ~nk = ~Rk/Rk is the normalized position vector of the nuclear spin. In the case

of an s-type wave function the anisotropic hyperfine interaction is much smaller

than the isotropic hyperfine interaction. Conversely, for p-type wave functions

the anisotropic dominates over the isotropic hyperfine interaction. Theoretical pre-

dictions, based on the anisotropic hyperfine interaction, estimated the interaction

strength of a QD hole spin with nuclear spins to be one order of magnitude less than

for a QD electron spin [19], which was experimentally confirmed [44].


2.2.4. Electron Spin Decoherence

In addition to an external applied magnetic field ~Bext, the QD electron spin expe-

riences a nuclear magnetic field ~Bnuc. Therefore spin precession about the vector of

the total magnetic field is expected ( ~Btot = ~Bext+ ~Bnuc). Especially the longitudinal

component Bznuc, i. e. , the component oriented parallel or opposed to ~Bext, changes

the precession frequency independent of the strength of the external field. If ~Bnuc

would be fixed and precisely known, the resulting phase shift could be exactly calcu-

lated. This is not the case in a QD; the nuclear field, generated by N ∼ 105 nuclear

spins, has a random and unknown value and orientation. Hence, the electron spin

picks up a random phase and decoherence occurs. For an unprepared nuclear spin

state, a Gaussian distribution of the nuclear field values Bznuc with a standard de-

viation of√〈(Bz

nuc)2〉 ∼

√N can be assumed, leading to a nuclear field fluctuation

of ∆Bnuc ' A√NgelµB

' 20− 40mT. [59] The dephasing of the electron spin is of the

form exp[−t2/(T ∗2 )2] with

T ∗2 =~√

2

gµB√〈(Bz

nuc)2〉

(2.16)

This leads to a typical dephasing time of T ∗2 ∼ 1ns [21]. Experimentally, the

T ∗2 time can be determined with a Ramsey experiment, which works as follows.

Assuming the external magnetic field is applied along the z-axis, the electron

spin is first rotated into the x-y plane. After a spin precession for a certain

time interval, the electron spin is rotated again by an angle of π and measured

along the z-axis. Because of the different phases picked up during the precession,

the measured signal is oscillating as a function of the chosen time interval. The

dephasing can be measured by the decay of the oscillation amplitude (see chapter 5).

For determining T ∗2 an average over several experiments with different Bnuc is

taken. Due to the nuclear field fluctuations a short coherence time can be observed.

The situation completely changes, if only one experiment with a single spin is

considered. Obviously the time scale T2 on which the phase of the electron spin is

lost would be longer than the previously considered value T ∗2 . Experimentally, the

random spin evolution during a certain time interval τ can be reversed by applying

a 180 rotation pulse (spin-echo pulse) and letting the spin evolve for a second

time interval of the same duration τ [60]. It is an important requirement that the

time constant of the nuclear field fluctuations is long compared to the spin echo

sequence. Therefore the effect of the random nuclear field can only be partially

eliminated, resulting in a spin echo T2 time. For simplification the spin echo T2

time will be referred to as T2 time in this thesis. Recent spin echo measurements on

QDs determined a value of T2 ∼ 3µs [61]. Since this decoherence is not irreversible,

with more advanced pulse techniques even longer coherence times exceeding 200µs


can be measured [10].

After looking into the dephasing mechanisms of the QD electron spin, we now focus

on electron spin flips caused by the nuclear field. For Bext Bx,ynuc electron-nuclear

spin flip-flops are allowed and lead to electron spin relaxation times of T1 ∼ 10ns.

As Bext increases, direct electron nuclear flip-flops become suppressed and T1

rapidly increases. At high magnetic fields T1 decreases again due to interaction

with acoustic phonons [9].

Experimental Methods 21

3. Experimental Methods

In this chapter an overview of the different spectroscopy techniques used to

investigate InGaAs QDs is given. While Photoluminescence (PL) is mainly

applied for a first characterization of the QDs, resonant spectroscopy tech-

niques namely resonant Rayleigh scattering and resonance fluorescence al-

low for a direct access to the optical transitions in the QD. Furthermore,

the development of a pump probe method that allows for a resonant mea-

surement of the Overhauser shift is described.

3.1. Photoluminescence

Photoluminescence (PL) can be applied to optically characterize charge states of

single QDs [30, 35, 62]. After exciting the QD sample with a non-resonant laser

above the band gap of GaAs, electron hole pairs are formed which diffuse near the

vicinity of the QD trap potential. Subsequently the exciton is trapped by the QD

potential and a relaxation to the QD ground state takes place by phonon scattering

on ps timescale [63]. Finally, the exciton recombines spontaneously after its lifetime

τ ∼ 1ns [64] and a photon is emitted.

Non-resonant excitation allows to excite different charge states of the QD which

can be spectrally resolved as a function of the gate voltage, leading to a charging

diagram as shown in Fig. 2.5. While PL is advantageous to achieve an overview of

the exciton energies of a particular QD, a drawback of non-resonant excitation is

the charging of defects in the QD environment. This can lead to shifted or broad-

ened transition energies compared to resonant measurements [65]. Additionally, the

spectral resolution is constricted and a lifetime limited line width of ~/τ ∼ 1µeV

cannot be resolved with a standard spectrometer. This problem can be overcome

with an additional Fabry-Perot cavity filter which suffers on the other hand from a

low photon throughput [66]. The polarization conservation of a PL experiment is

strongly dependent on the laser excitation energy. A non-resonant excitation to the

p-shell states of the QD (referred to as photoluminescence excitation (PLE) ) can

lead to a maximal polarization preservation of ∼ 80%. This plays an important role

in polarization of nuclear spins which will be explained in detail in section 3.4.1.

22 Experimental Methods

3.2. Resonant Rayleigh Scattering

Besides the limited spectral resolution in non-resonant experiments an additional

reason to perform resonant spectroscopy is the conservation of the coherence during

the optical excitation process. To perform resonant spectroscopy a narrow band

laser is scanned across the excitonic resonance. Subsequently, the intensity of the

transmitted or reflected light is detected with a photodiode. The measured intensity

is given by

Idet ∝ 〈| EL(xdet) + EQD(xdet) |2〉= 〈| EL(xdet) |2〉+ 2〈| EL(xdet)EQD(xdet) |〉+ 〈| EQD(xdet) |2〉, (3.1)

where EL(xdet) is the electric field of the laser and EQD(xdet) is the electric field of the

QD at the detector. The time averaging during the measurement is represented with

brackets. Due to the stabilized laser intensity, the first term results in a constant

offset to the signal. By considering the fact that | EQD || EL | the third term

in (3.1) can be neglected. Therefore the measured intensity is given by

Idet ∝ 〈| EL(xdet)EQD(xdet) |〉. (3.2)

The response of the QD to the incident laser field is given by EQD = χ(δ)EL(0),

where χ(δ) is the susceptibility as a function of the laser detuning δ and EL(0)

describes the laser field at the position of the QD. For a complex susceptibility χ(δ) =

χ′(δ) + iχ′′(δ) the first term relates to the dispersion while the second term stands

for the absorption. By simplifying the QD to a two level system, the susceptibility

can be expressed as follows

χ′(δ) ∝ γδ

δ2 + γ2χ′′(δ) ∝ γ2

δ2 + γ2, (3.3)

with the radiative damping rate γ = Γ/2 and the laser detuning δ = ω0−ωL between

the laser energy ~ωL and the excitonic transition energy ~ω0. Additionally, one has

to take into account the phase shift of π while the laser beam is propagating through

its focus, the Gouy phase [67, 68]. For this reason the laser field is phase shifted by

π/2 compared to the electric field of the scattered light by the QD. This leads to a

detected intensity given by

Idet ∝ Re〈E2L(0)eiπ/2(χ′ + iχ′′)〉. (3.4)

Due to the fact that only the absorptive part of the QD response is in phase with

the laser field at the detector, its interference with the incoming field is measured.

3.2. Resonant Rayleigh Scattering 23

Experimentally, the measured quantity in a differential transmission experiment is

the transmission contrast dT/T , which is given by

dT

T∝ σ0

A

γ2

δ2 + γ2, (3.5)

where A corresponds to the spot size of the laser and σ0 is the scattering cross

section. Alternatively, the reflection signal dR/R can be investigated since the QD

is emitting light in all spatial directions. A detailed derivation of equation (3.5) is

given by Karrai et al. [69]. For simplicity differential transmission and reflection

are referred as dT and dR, respectively. The measured dT signal as a function of

the laser detuning leads to a Lorentzian signal as shown in Fig. 3.1. The slight

asymmetry in the Lorentzian can be explained with an additional small component

from the dispersion. Such a situation occurs if the emitted light from the QD

experiences an additional phase shift due to scattering effects at the sample surface

or the gate.

Resonant Rayleigh scattering revealed the two-level nature of a QD that manifests

in power broadening and a saturation of the signal [70]. With an increase of the

laser power, the incoherent part of EQD increases and therefore the measurement

contrast decreases resulting in a saturation of the signal.

A particularity of the resonant measuring method differential transmission

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

dT

/T (

%)

Detuning (eV)

Figure 3.1.: dT/T measurement for X−. Typical dT/T signal versus detuningfor X− below saturation. The optical line width is 1.7µeV.

(dT) is the challenge to resolve a change in the reflection of ∼ 0.1%. Usually, an

increase of the signal to noise ratio (SNR) is achieved by using Lock-In technique.

To perform a Lock-In measurement, the resonance energy of the exciton needs to be

modulated. This is realized with the quantum confined Stark effect, which allows


to control the exciton transition energy via the gate voltage (see chapter 2.1.4).

To resolve the true line shape of the resonance, a square wave modulation of

Vpp = 100mV, which is an order of magnitude larger than the line width of

the resonance, is applied to the gate. Therefore two replica of the resonance at

V− = Vr− 1/2Vpp and V+ = Vr + 1/2Vpp can be observed in the measured spectrum.

In a Lock-In detection only the first Fourier component of the modulation frequency

is integrated and all other frequencies are filtered out. Therefore a higher SNR can

be achieved [31, 71].

3.3. Resonance Fluorescence

An alternative approach to perform resonant spectroscopy is to measure di-

rectly the response of the QD to the incident laser field, which corresponds to

〈| EQD(xdet) |2〉 in equation 3.1. The main difficulty is to achieve a sufficient

suppression of the laser such that the resonance fluorescence photons (RF) from

the QD are not overwhelmed by the laser background. The approach by Muller

et al. [72] relies on a spatial filtering of the resonant laser. Alternatively, a polar-

ization suppression of the laser by a factor of 106 allows for measuring resonance

fluorescence [73, 74]. Experimentally, this is realized by operating the microscope

in a dark-field configuration [75]. A linear polarizer that is perpendicular to the

incoming linearly polarized laser field is placed in front of the collection fiber. Sub-

sequently the RF photons are sent to an avalanche photodiode (APD). In contrast

to a dT measurement where the polarization of the signal is given by the laser, only

the photons that have a polarization component orthogonal to the excitation laser

are collected in RF measurements. Typically 103 − 106 counts/sec. are detected

from a single QD.

3.4. Pump Probe Measuring Methods

To explore the electron nuclear spin system with increasing transverse magnetic

field in chapter 4, it would be ideal to directly observe the dynamics of the electron

and nuclear spins. However, only the electron spin is directly accessible with current

measurement techniques. In contrast, nuclear spin polarization can only indirectly

be measured via the electron spin. This is possible due to the hyperfine interaction

that generates an energy shift (Overhauser shift ∆Eos) on the electron spin levels

which is proportional to the nuclear spin polarization. In this section we describe

a method to efficiently polarize nuclear spins and optically measure the magnitude

of nuclear spin polarization. Furthermore, two different pump probe measuring

methods are developed to access ∆Eos with resonant spectroscopy.

3.4. Pump Probe Measuring Methods 25

3.4.1. Optical detection of Nuclear Spin Polarization

There exist several methods to polarize nuclear spins in an InGaAs QD [66, 76, 77].

Most of them relay on the scalar form Se · I of the hyperfine interaction that con-

serves the total spin and the energy. By the relaxation of an electron spin via this

interaction, the spin angular momentum is transferred to a nuclear spin in the

so-called flip-flop process. Therefore, a large electron spin polarization obviously

helps to create significant nuclear spin polarization. Due to the selection rules

in a QD, which correlate the electron spin states with the light polarization, a

large electron spin polarization can be achieved by optical means. Since a sizable

nuclear spin polarization at Bx = 0 is required for the investigation of nuclear spin

dynamics in a transverse magnetic field Bx, we polarize nuclear spins by exciting

the QD into one of its excited p-shell states [66].

To guarantee a sizable nuclear spin polarization a single electron spin has to be

s- s+

ΔEOS

p-shell

s-shell s+

-100 0 1000

20000

40000

s+/s

+

s+/s

-

PLE

count

rate

(cts

/0.2

s)

Detuning (eV)

EOS

=18eV

(a) (b)

Figure 3.2.: Optical detection of nuclear spin polarization in X+. (a) Nu-clear spin polarization is achieved by resonantly pumping a single QD to one ofits excited states in the p-shell with a circularly polarized laser (e.g. σ+, green ar-row). Subsequently the emitted co- and cross- polarized photons are detected. (b)PLE spectra of X+ in zero magnetic field measured under co-polarized (σ+/σ+) andcross- polarized (σ+/σ−) excitation/detection configuration. The energy splitting∆Eos is induced by the Overhauser field.

either in the ground or excited state of the exciton, which is only fulfilled in a

charged QD. Additionally, we would like to monitor the electron spin polarization

via the polarization of the fluorescence, therefore the electron spin has to be in the

excited state. Out of these reasons we will focus our studies on the X+ state. A

schematic drawing of the X+ energy diagram is shown in Fig. 3.2 (b). The QD is

excited into the p-shell with a σ+ polarized laser. The Subsequent relaxation to

the s-shell takes place within picosecond timescales. The relaxation occurs due to a

combination of phonon-mediated relaxation and co-tunneling through the electron


reservoir. Finally the X+ state recombines to |⇓> (|⇑>) under the emission of a σ+

(σ−) polarized photon. These photons can be analyzed with a spectrometer. The

polarization configuration of the p-shell laser and the emitted photons is denoted

(σα/σβ) where σα is the excitation polarization and σβ represents the detection

polarization. Due to nuclear spin polarization an energy shift between |⇓⇑↓>and |⇑⇓↑> occurs. This so-called Overhauser shift ∆Eos is proportional to the

magnitude of the nuclear spin polarization in z-direction and can be measured by

separately analyzing the co- and cross- polarized emitted photons. Such a polariza-

tion sensitive detection method allows for a high spectral resolution up to 4µeV. For

all measurements the p-shell state was excited to saturation, which results in a high

polarization preservation. The polarization preservation can be quantified with the

degree of circular polarization which is defined as ρ+ = (I+− I+)/(I+ + I−), where

Ix is the PLE intensity in the (σ+, σx) polarization configuration. Experimentally,

a polarization preservation of up to ρ+ ∼ 80% can be measured.

The emitted PLE photons as a function of the energy detuning are shown for co-

and cross- polarized detection with respect to the excitation polarization in Fig. 3.2

(a). Clearly a high degree of polarization preservation is exhibited. The energy shift

of ∆EOS = 18µeV between the two polarization configurations corresponds to an

effective magnetic field experienced by the electron of Beff = ~ωeOS/geµB ∼ 500mT.

In summary, resonant excitation into the p-shell allows to polarize nuclear spins. In

order to measure the nuclear spin polarization the energy splitting of co- and cross-

polarized detection is extracted. Additionally, in X+ the electron spin polarization

can be measured determining the polarization of the emitted photons.

To achieve a higher spectral resolution for measuring nuclear spin polarization

compared to a spectrometer and to have a coherent measuring method to directly

investigate the optical transitions, resonant spectroscopy techniques in combination

with p-shell excitation were investigated. In this case the limiting factors for the

resolution are given by the optical line width that is determined by the homogeneous

broadening and the inhomogeneous broadening that arises from charge fluctuations

and distortions of the optical line shape.

3.4.2. Pump Probe Method with Resonant Rayleigh Scattering

One possibility to access the Overhauser shift resonantly is to combine the p-shell

pumping with resonant Rayleigh scattering, namely dR, which is described in

section 3.2.

We performed a dR measurement on X+ for different p-shell laser powers. The

results of this measurements are shown in Fig. 3.3 (black). Obviously, there is an

ideal PLE laser power of 12µW to achieve the largest reflection contrast. The signal


strength in resonant absorption measurements, like dR, is strongly dependent on

the population of the ground state of the optical transition [15, 78]. Only with a

sizable population of the ground state, a dR signal can be measured. Since the

ground state of the X+ consists of a single hole inside the QD, the PLE laser is

populating the QD with holes which is explained in detail in chapter 2. For weaker

PLE laser power than 12µW the reflection contrast is smaller due to reduced

population of the X+ ground state. A larger PLE laser power than 12µW results

in a population of excited states of X+. Therefore the single hole population inside

the QD is reduced, which leads to a smaller reflection contrast.

To investigate nuclear spin polarization in a transverse magnetic field (see chap-

1 1 0 1 0 00 . 0 3 50 . 0 4 00 . 0 4 50 . 0 5 00 . 0 5 50 . 0 6 00 . 0 6 5

L a s e r p o w e r ( µW )

Refle

ction

contr

ast (%

)

0

5

1 0

1 5

2 0

∆EOS (µeV)

Figure 3.3.: Comparison between dR and ∆ EOS. Reflection contrast (black)and ∆EOS (blue) as a function of PLE laser power.

ter 4) we are interested in a large Overhauser shift ∆Eos in addition to a sizable

reflection contrast. A comparison of the reflection contrast (black) and ∆Eos (blue)

as a function of the PLE laser power is shown in Fig. 3.3. Sizable nuclear spin

polarization of ∆Eos = 19µeV can only be achieved for a large PLE laser power of

several 100µW, where the reflection contrast is small.

One possibility is to choose the p-shell laser power such that a sizable reflection

contrast is reached at the cost of a reduced nuclear spin polarization. Another

approach is to combine the large Overhauser shift with a sizable reflection contrast

in a pulsed pump probe technique. While nuclear spins are polarized in the pump

sequence, a reduced p-shell laser power enables a sizable reflection contrast for the

resonant measurements in the probe sequence. To polarize nuclear spins, a p-shell

laser power of several hundred µW is needed. In order to perturb the system as

little as possible, the dR measurements take place with an additional weak p-shell

laser that is needed to charge the QD with a single hole. The probe signal depends

dramatically on the time constants of the experiment. If the measurement time


1 10 100

0

2

4

6

8

10

decay

=66ms

E

os(

eV

)

Twait

(ms)

0.1 1 10

0

2

4

6

8

10

E

OS

(eV

)

Tpump

(ms)

buildup

=2ms

(c) (d)

shutter

sample

t

t

c

o

AO

M

c

o

shutt

er

Tpump Tprobe Twait

(a) (b)

Figure 3.4.: Nuclear spin dynamics measured with p-shell emission. (a)Setup for pump probe measurement. The p-shell excitation laser is pulsed withan AOM. The pump laser is blocked with a mechanical shutter in front of thespectrometer. (b) Pulse sequence for the pump probe experiment. A long pumppulse Tpump polarizes nuclear spins, after a waiting time Twait a short p-shell laser(Tprobe) is needed to probe the sample. (c) ∆Eos is measured as a function ofTpump. An exponential fit (red) leads to a buildup time of nuclear spin polarizationτbuildup =2ms. (d) ∆Eos is measured as a function of Twait. An exponential fit (red)leads to a decay time of nuclear spin polarization τdecay =66ms.

(Tprobe) is larger than the decay time of nuclear spins (τdecay), ∆Eos is vanishes and

only one resonance is measured with a linearly polarized probe laser. In the case of

Tprobe < τdecay the Overhauser splitting can be determined with the energy splitting

between the two circularly polarized resonances. Hence this technique allows access

to the nuclear spin decay time by varying Tprobe.

In all following experiments we rely on the detection of the maximal Overhauser

shift. Therefore the time constants for build up and decay of nuclear spin polariza-

tion in X+ have to be determined. We realized this by a pump probe experiment

based on PLE as follows. In order to achieve well defined pump and probe pulses,

the cw laser amplitude is modulated with an AOM. Furthermore a mechanical

shutter is used to block the pump pulses while allowing the probe pulses to reach


the spectrometer. A schematic picture of the pulse sequences is shown in Fig. 3.4.

For measuring the buildup (decay) of nuclear spin polarization, Tpump (Twait) are

varied while keeping all other parameters fixed. To improve the signal to noise

ratio, the pump-probe sequences are repeated several times during the measurement

process, while the signal is accumulated on the spectrometer CCD.

In Fig. 3.4 (c) and (d) the resulting buildup and decay curves for nuclear spin

polarization in X+ are shown. Fitting the data with an exponential curve gives a

buildup time of τbuildup = 2ms and a decay time of τdecay = 66ms.

While the buildup time is determined by the method which is used to polarize

nuclear spins, the decay time is a characteristic of the QD state. Similar to previous

experiments performed by Maletinsky et al. [8] a buildup time of several ms is

measured. Since for X+ there is a single hole in the ground state, the fast electron

mediated nuclear spin decay should be strongly suppressed [8, 79]. Therefore a long

nuclear spin decay time for X+ is expected and was measured to be on the order of

seconds [8]. We find a much shorter τdecay of 66ms, which is most likely due to less

well defined charge state compared to the experiments performed by Maletinsky et

al. [8].

Not only for the nuclear spin polarization but also for the signal to noise ratio (SNR)

of the probe signal the timescales of a pump probe measurement are important.

The ratio between Tprobe and Tpump determines the scaling factor for the integration

time which is needed to arrive at the same SNR as for a cw experiment. Hence

the measured τdecay =66ms in combination with a short buildup time τbuildup =2ms

should result in a reasonable signal to noise ratio.

Usually, differential reflection dR is measured with a Lock-In technique (see

section 3.2). Therefore a closer look at the time constants of the Lock-In amplifier

is needed for performing a resonant pump probe measurement in combination with

dR. A commercially available Lock-In amplifier has a time constant Tc that can be

chosen between several hundred µs up to seconds. Obviously a larger Tc results in

a better signal to noise ratio. Considering a change of the input signal many Tc are

needed to reach the output signal. This is due to the response time of the low-pass

filter. The timing and synchronization of the pump probe pulses is controlled

with a data acquisition (DAQ) device based on a clock period of 2µs. Because

of its finite memory the modulation sequences are restricted to ∼ 2s. During the

waiting time between the modulation sequences the feedback loop of the EOM

intensity modulator is performed and results in a new pulse sequence which has

to be written on the DAQ device. The combination of the waiting time between

the modulation sequences with the response time of the Lock-In amplifier (τLock-In)

leads to a substantial amount of time where no signal can be acquired (Fig. 3.5).

To circumvent this problem we tested an alternative approach consisting of a


0 1 2 30

1

0 . 05 . 0 x 1 0 - 6

1 . 0 x 1 0 - 5

1 . 5 x 1 0 - 5

01 0 02 0 03 0 0

0 1 2 3

Volta

ge (V

)

T i m e ( s )

m o d u l a t i o ns i g n a lτL o c k - I n

dR (1

E-8A

) ( b )

( c )

Ωpu

mp (µ

W) ( a )

Figure 3.5.: Measurement scheme for pump probe with dR. (a) Measure-ment of the pump laser power Ωpump versus time. The gap between the modulationsequences arises from writing the sequence on the DAQ device memory. (b) Cor-responding dR signal from Lock-In amplifier versus time. The strong pump laserbetween the modulation sequences leads to an increase of the dR signal. (c) Digitalsignal to be multiplied with Lock-in signal. To avoid memory effects the modulationof the dR signal is shifted by τLock-In.

software based Lock-In amplifier, which is implemented directly on the DAQ

device. With this method a synchronization of the Lock-In amplifier with the

pump probe pulses is possible such that the response time of the Lock-In amplifier

can be reduced. To compare the commercially available Lock-In amplifier from

Signal Recovery (SR) with the software based Lock-In amplifier, a cw experiment

was performed. The result is shown in Fig. 3.6. For the software based Lock-In

amplifier a decrease of the signal to noise ratio by a factor of ten is exhibited, which

is caused by the finite bandwidth of the used digital to analog converter on the

DAQ device. Even though a faster response time can be reached with a Lock-In

amplifier implemented on the DAQ device compared to the SR Lock-In amplifier,

the combination with the strongly reduced signal to noise ratio will not result in

a better performance for the pump probe experiment. Therefore the pump probe

experiments were realized with a commercially available Lock-In amplifier, taking


- 0 . 1 0 - 0 . 0 8 - 0 . 0 6 - 0 . 0 4- 5 . 0 x 1 0 - 5

0 . 0

5 . 0 x 1 0 - 5

dR (1

E-8A

)

G a t e V o l t a g e ( V )

D A Q c a r d S R L o c k - I n

Figure 3.6.: dR measurement with software based Lock-In amplifier. dRmeasurements as a function of gate voltage performed with a SR Lock-In amplifier(black) and a software based Lock-In amplifier (DAQ device) (red).

into account the slower response time compared to a Lock-In amplifier which is

based on the DAQ device.

For a first attempt at pump probe measurement with dR only the σ+ polarized

p-shell laser is pulsed while the resonant laser and a hole generating weak PL laser

is continuously on. During the entire measuring process, the gate is modulated for

the Lock-In detection (see Fig. (b)). In this experiment a fiber based EOM intensity

modulator served as a fast switch of the p-shell laser intensity. Besides the faster

rise time compared to an AOM, the fiber based intensity modulator has a larger

throughput. The intensity modulation is achieved with two EOMs that are forming

a Mach-Zehnder interferometer. By applying a voltage to the EOM the phase of the

light in one interferometer arm can be shifted with respect to the other. If the light

after recombination is out of phase a destructive interference is observed, resulting

in a reduced intensity. Unfortunately the voltage needed to apply to the EOM

for achieving a maximal suppression is drifting due to temperature fluctuations.

Either the environmental temperature or the heating of the device with a strong

pump laser can lead to temperature changes inside the intensity modulator. These

changes are rather slow therefore the voltage needed for a maximal suppression can

be stabilized with a control loop feedback mechanism (PID).

Similar to the co- and cross- circularly polarized probing performed in non-resonant

measurements, we compare σ+ and σ− resonant absorption. In Fig. 3.7 (a) dR

signal for both polarizations is plotted as a function of the laser detuning. From

top to bottom the measuring time between the pump pulses which is indicated with

Twait is decreased. Clearly, there is an observable energy shift up to 5µeV between

σ+/σ+ and σ+/σ−. Additionally, the asymmetry increases as the ratio between

Tpump and Twait increases which can be explained as follows. The consequence


-10 -5 0 5

-0.1

0.0

0.1

-0.1

0.0

0.1

0.2

-0.1

0.0

0.1

0.2

0.3

0.4

-10 -5 0 5

dR

(V

)

Detuning (eV)

dR

(V

)

Tpump

=10ms,

Twait

=3ms

Tpump

=10ms,

Twait

=10ms

Tpump

=10ms,

Twait

=5ms

+/

+

+/

-

dR

(V

)

- +

ΔEOS

z

z

z

p-shell

s-shell +

Gain

(a)

(c)

t

t

c

o

EO

M

off

resonant

on

gate

Tpump Twait

(b)

Figure 3.7.: Pump probe measurement with dR. (a) Pump Probe measure-ments of the Overhauser shift for different polarizations of the resonant laser. DRsignal versus detuning for a resonant σ+ polarized laser is shown in black, while for aresonant σ− polarized laser it is shown in red. The waiting time Twait and thereforethe measurement time is decreasing from top to bottom. (b) Measurement scheme.The pump laser is pulsed with an EOM, while the resonant laser and a weak PLlaser (to generate the hole for X+) is continuously on. Gate voltage is modulatedduring the measurement process and even on resonance during Tpump. (c) Energylevel diagram for X+. The σ+ pump laser polarizes nuclear spins by exciting to thep-shell (shown in green). Resonant measurements can be performed with σ+ (redarrow) or σ− (blue arrow). A large population of |⇓⇑↓> by the pump laser leads togain (indicated with dotted orange arrow).

of a strong p-shell laser is a large population of the |⇓⇑↓> state which leads

to stimulated emission (gain) in a dR measurement (Fig. 3.7 (c)). Therefore a

negative dR signal can be observed for a σ+ polarized resonant laser. Especially

the dependence on the ratio between Tpump and Twait can be understood by the

fact that the large population in |⇓⇑↓> is increasingly dominating for shorter Twait,

leading to the observed stimulated emission. In order to extract the energy splitting

∆Eos between σ+ and σ− polarized resonant excitation, the gain process is avoided

by modulating the gate voltage away from the X+ resonance during the strong

pump pulse.


- 5 0 5 1 0 1 5- 0 . 0 20 . 0 00 . 0 20 . 0 40 . 0 6

0 . 0 0

0 . 0 5

0 . 1 0

0 . 0

0 . 1

0 . 2- 5 0 5 1 0 1 5

dR (V

)

D e t u n i n g ( µe V )

T p u m p = 1 2 m s ,T w a i t = 1 . 4 m s

T p u m p = 1 2 m s ,T w a i t = 0 . 7 m s

dR (V

)

( b )

( c )

T p u m p = 1 2 m s ,T w a i t = 3 . 5 m s

dR (V

)

( a )

Figure 3.8.: Pump probe measurement with dR for different Twait. dR sig-nal versus detuning for a linearly polarized resonant laser is shown . The waiting timeand therefore the measurement time is decreasing from top to bottom. (a)With awaiting time of 3.5ms, nuclear spin polarization is already decayed to ∆EOS=3.5µeV.(b), (c) A further decrease of the waiting time leads to ∆EOS=7µeV.

The high spectral resolution of dR compared to PLE allows to measure the

Overhauser shift directly in a linear basis. In Fig. 3.8 the dR signal as a function

of the laser detuning is shown for decreasing measurement time Twait between the

pump pulses from top to bottom. With the realization of a gate modulation away

from the X+ resonance during the strong pump laser, two separated resonances

with a Lorentzian dR signal can be measured for a linearly polarized probe laser.

The maximal nuclear spin polarization is reached for Twait = 1.4ms. Because of

the constant pump time Tpump the ratio between the measuring time and Tpump is

decreasing from top to bottom and thus the signal to noise ratio (SNR).

Such a short nuclear spin depolarization time is surprising, especially since previous

measurements exhibited a τdecay of 66ms. One reason for a shortened decay time

could be the influence of the weak PL laser or the probe laser which are continuously


on. For a better understanding a detailed study including a pulsed weak PL laser

and probe laser would be needed.

3.4.3. Pump probe Method with Resonance Fluorescence

Although a resonant measurement of the Overhauser shift can be realized with

a pump probe method based on dR, the need for a Lock-In technique leads to a

substantial amount of time when no signal can be acquired as discussed in the

previous section. Alternatively resonance fluorescence (RF), that is explained in

detail in section 3.3, can be used as a resonant measuring method. This has the

advantage that no Lock-In technique is needed and therefore a higher SNR in

a pump probe experiment should be achieved compared to dR. Switching from

resonant Rayleigh scattering to RF does not alter the experiment substantially, only

changing the polarization of the excitation laser is a much more time consuming

task for RF than for dR because of the alignment of the polarizer in order to achieve

a high polarization suppression.

In Fig. 3.9 (a) a scheme for the pump probe measurements with RF is shown.

Similar to the previous pump probe technique, the pump pulses are generated

with an EOM while a weak hole generating PL laser and the resonant laser

are continuously on. Especially during Tpump a suppression of the background

is difficult because the emitted PLE photons cannot be distinguished from RF

photons. Therefore only the photons between two pump pulses, during Twait are

detected. The resulting RF count rate for a σ− polarized pump laser and a linearly

polarized resonant laser as a function of the laser detuning is shown in Fig. 3.9 (b).

Compared to the measurement shown in Fig. 3.8 a reasonable signal to noise ratio

is achieved with less integration time.

To conclude, a resonant pump probe technique to measure nuclear spin po-

larization is realized with dR and RF. A comparison of these methods shows an

easier polarization control for dR which is contrasted by the improved signal to

noise ratio of RF measurements.


t

t

c

o

EO

M

off

on

Measure

RF

Tpump Twait

(a) (b)

-5 0 5 100.05

0.10

0.15

0.20

0.25

0.30 -/ lin

RF

count

rate

(cts

/0.2s)

Detuning (eV)

Figure 3.9.: Pump probe measurements with RF. (a) Scheme for a pumpprobe experiment measuring the RF. The pump laser is pulsed with an EOM, whilethe probe laser and the weak hole generating PL laser are continuously on. The RFphotons are only counted during the time period Twait, when the strong pump laseris off. (b) RF pump probe measurement as a function of the laser detuning. Thepump laser is σ− polarized, while the resonant laser is linearly polarized.

Nuclear Spin Polarization in Transverse Magnetic Fields 37

4. Nuclear Spin Polarization in

Transverse Magnetic Fields

We investigated nuclear spin polarization in individual self-assembled

InAs/GaAs quantum dots for different sample structures in transverse mag-

netic fields. Hanle depolarization curves are measured to explore nuclear

spin dynamics, leading to an anomalous Hanle curve with a broadening up

to 1T. Even though nuclear spin polarization is comparable for different

sample structures at zero magnetic field, the broadening of the anomalous

Hanle curve is substantially different. The critical field, at which the pho-

tocreated electron spin polarization breaks down, depends strongly on the

electron spin lifetime. For further investigation of direction and strength of

the Overhauser field causing the broadening of the Hanle curve, a resonant

measurement technique is applied.

4.1. Introduction

A phase transition is the transformation of a physical system from one state

to another, such as vaporization and condensation. These transitions manifest in

unconventional and unexpected physical behavior and do not only occur at finite

temperature. Even at zero temperature or ~ω > kBT quantum fluctuations (at

energy ~ω) are expected to lead to so-called quantum phase transitions. In order

to have experimental access to a quantum system, a coupling to the environment

is needed which leads to dissipation. Dissipative phase transitions can occur if an

environmental parameter is varied such as a magnetic field or an external coherent

source (e. g. a laser) [80–82]. Investigations on dissipative phase transitions (DPT)

are expected to lead to a deeper understanding of the unexpected physical behavior

of systems at low temperatures.

A model exhibiting first- and second- order DPT is the central spin system which

is externally driven and decays through interaction with a Markovian environ-

ment [83]. In its simplest form, a central spin system consists of a set of spin-

1/2 particles, uniformly coupled to a single spin-1/2. A possible experimental

38 Nuclear Spin Polarization in Transverse Magnetic Fields

implementation of such a system is a negatively charged InGaAs QD, where nuclear

spins represent the set of spin- 1/2 particles and the electron is the single spin-1/2

particle. Accordingly, the nuclear spin ensemble is expected to undergo phase

transitions. The behavior of the nuclear spin ensemble can be monitored via the

interaction between nuclear and electron spins. If a negatively charged InGaAs QD

is driven by a laser, it experiences spontaneous decay and therefore satisfies the

requirements of the theoretical model studied by Kessler et al. [83]. On the other

hand the model does not consider inhomogeneous hyperfine coupling, the electron

Zeeman splitting and the nuclear quadrupolar interactions [51, 52] in InGaAs QDs,

which raises the question to what extent an experimental observation of dissipative

phase transitions is possible in this system. Still, in recent experiments on InGaAs

QDs indications for a first order phase transition in the nuclear spin ensemble

were observed, including optical bistability as a function of a transverse magnetic

field Bx [20]. Therefore we want to further investigate the electron - nuclear spin

system under the influence of Bx with an increased spectral resolution to identify

discontinuities in the polarization of the nuclear spin ensemble. An abrupt change

in electron and nuclear spin polarization would signal a dissipative phase transitions

and lead to new insights into the electron nuclear spin system.

4.2. Hanle experiment

In a Hanle measurement, the dynamics of an electron spin in a transverse mag-

netic field Bx is determined by optical means. In the course of such an experiment

an exciton is created with a circularly polarized excitation laser. A schematic draw-

ing of the measurement geometry is shown in Fig. 4.1(a). Because of the selection

rules in a QD, the electron spin polarization S0 is optically defined along the z-axis

parallel to the Poynting vector of the excitation laser. In the case a finite magnetic

field Bx is applied, the optically generated electron spin undergoes Larmor preces-

sion around the x-axis at frequency Ωel = gelµBB/~ (see Fig. 4.1(b)). Due to the

fact that the projection of the electron spin onto the z-axis defines the degree of

circular polarization of the emitted photons, a depolarization of the PL light can

be observed as a function of Bx. This dependence allows to determine both, the

electron spin lifetime τs and the recombination time τ of the exciton [48].

The electron spin precession together with spin pumping, spin relaxation and re-

combination can be described with the following equation of motion for the spin

vector ~S :d~S

dt= ~Ωel × ~S −

~S

τs−~S − ~S0

τ. (4.1)

4.3. Anomalous Hanle Effect 39

B=0T

y

z

Bx

S

Bx

y

z

x

(a)

(b)

Figure 4.1.: Scheme illustrating Hanle measurements. (a) An external mag-netic field Bx is applied transverse to the Poynting vector of the i.e. σ+ polarizedlaser (orange arrow). Electron spin dynamics are investigated with a polarizationanalysis of the emitted light (black arrow). (b) The electron spin polarization is op-tically defined along the z-axis. For finite Bx the electron spin (red arrow) precessesaround the x-axis leading to a decay of the polarization along the z-axis.

The first term represents the spin precession in the magnetic field, the second term

describes the spin relaxation and the third term denotes the generation of the spin

by optical excitation (S0/τ) and its decay (−S/τ). In steady state (d~Sdt

= 0) and in

the presence of a magnetic field transverse to S0 (~Ωel ⊥ ~S0) one obtains

Sz(B) =Sz(0)

1 + (Ωτ ∗)2, (4.2)

with the effective time τ ∗ = (τ−1 + τ−1s )−1. Hence, a Lorentzian with a half width

of

B1/2 = (Ωelτ∗)−1 = ~/(gelµBτ ∗) (4.3)

results for the spin projection Sz and therefore the degree of circular polarization of

the emitted light as a function of Bx.

4.3. Anomalous Hanle Effect

Running a Hanle experiment is one of the few techniques to monitor the behavior

of nuclear spin polarization in transverse magnetic fields. Theoretically, the behav-

ior of nuclear spins should closely resemble the one of electron spins. In a transverse

magnetic field Bx ≥ BKnight, nuclear spins should exhibit a Larmor precession, which

would then lead to a fast decay of the optically generated nuclear spin polarization.

In that case, the Hanle curve is expected to exhibit a Lorentzian line shape, as


described by Eq. 4.3, with a FWHM of 30mT for a ge = 0.5 and an effective time

dominated by the trion radiation lifetime τ ∗ ∼ τ = 0.7ns.

A typical Hanle depolarization curve, measured according to section 3.4.1, is shown

in Fig. 4.2 (a). The most striking anomaly to the theoretical predictions is the 40

times broadening of the FWHM. This unexpected broadening is termed anomalous

Hanle effect. In addition, the Hanle curve exhibits a large hysteresis of 0.5T. The

degree of circular polarization is continuously decreasing for magnetic fields up to

0.5T, where it reaches a minimum value of 50%. For larger magnetic fields the de-

gree of circular polarization increases again until a sudden drop down from 70% to

0% is observed at a critical magnetic field Bcrit = 1.1T. Similar results have been

presented in the detailed studies by Krebs et al. [20].

Together with the hysteresis, the broadening of the Hanle curve is a strong indication

for nuclear spin effects. By modulating σ+/σ− polarization at kHz frequency, an

optical polarization of nuclear spins can be prevented [8]. This effect was exploited

in an additional measurement. The resulting Hanle curve exhibited a narrowing.

Therefore we conclude that the nuclear spin polarization is the cause for the anoma-

lous broadening of the Hanle curve in InGaAs QD.

In addition to the PL circular depolarization data, the Overhauser shift can be ex-

tracted from the same experiment (Fig. 4.2 (b)). The results show a decrease of

nuclear spin polarization along the z-axis with Bx. At the same time a high degree

of PL circular polarization is measured (Fig. 4.2 (a)). This leads to the conclusion

that a nuclear spin polarization along the x-axis is very likely to compensate Bx

and prevents the electron spin from performing Larmor precession. The underlying

mechanism that could cause a rotation of the nuclear spin polarization from the z-

axis to the x-axis as a function of Bx is not fully understood. Such a situation could

result from a tilted effective magnetic field, that is stabilized along the z-axis by a

Bnucz due to nuclear quadrupolar interactions (QI) [20]. This is further supported

by the modest broadening of the Hanle curves measured on strain-free droplet QDs,

which do not exhibit QI [84]. Should there exist a tilted effective magnetic field, a

finite nuclear spin polarization along the z-axis must be observed up to Bcrit. The

data shown in Fig. 4.2 (b) do not allow to draw a clear conclusion in this respect.

To verify the existence of such a nuclear polarization component an experimental

technique with increased resolution needs to be applied.


-1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

PL c

ircula

r pola

rization

Bx(T)

(a)

(b)

-1.5 -1.0 -0.5 0.0 0.5 1.0

0

5

10

E

(eV

)

Bx(T)

Figure 4.2.: Anomalous Hanle effect. (a) Hanle depolarization curve as a func-tion of the transverse magnetic fieldBx measured on a 25nm tunneling barrier sample(sample B). Arrows indicate field sweep direction. (b) Corresponding energy shift∆EOS between co- and cross- polarized detection as a function of Bx.

4.4. Hanle measurements for samples with different

electron spin lifetimes

In the previous section, the Hanle effect was studied for a sample with 25nm

tunnel barrier (sample B) and exhibits a spectacular Hanle curve broadening

because of nuclear spin polarization. For a further investigation of the connection

between nuclear spin polarization and Bcrit we measured samples with different

electron spin lifetimes. This is of particular interest since the decay of nuclear spins

can be caused by a randomization of the electron spin [5]. Therefore the increased

electron spin lifetime of a 35nm tunnel barrier sample (sample A) compared to a

25nm tunnel barrier sample (sample B) should result in a better preservation of

nuclear spin polarization which could lead to an increased magnitude of nuclear

spin polarization and to even larger values for Bcrit.

In this section, we present a series of Hanle measurements for different values of

nuclear spin polarization at Bx = 0 on the same QD to verify to what extent this


-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

PL c

ircula

r pola

rization

Bx(T)

-1.0 -0.5 0.0 0.5 1.0

0

5

10

15

20

E

(eV

)

Bx(T)

B

B (a)

(b)

0.3 0.4 0.5 0.6 0.70

5

10

15

20

E

OS(

eV

)

Bcrit

(T)

(c)

Figure 4.3.: Hanle measurements for different nuclear spin polarizations.(a) Hanle depolarization curves for a QD on sample B with a 25nm tunnelingbarrier. Arrows indicate field sweep direction. (b) Corresponding energy splitting∆E between co- and cross- polarized detection. (c) Extracted Bcrit for different∆Eos at Bx = 0T.

parameter influences the broadening of the Hanle curve Bcrit. Subsequently, the

width of the Hanle curve will be compared for the two different sample structures

A and B.

A typical Hanle depolarization curve for sample B is shown in Fig. 4.3(a). Similar

to the measurements in the previous section, an increased broadening up to

Bcrit = 0.75T and a hysteresis can be observed. The corresponding nuclear spin

polarization in z-direction is extracted and plotted as a function of Bx (Fig. 4.3(b)).

At Bx = 0T a large Overhauser splitting of ∆Eos = 20µeV is obtained that

decreases continuously as a function of Bx until Bx = Bcrit. At this point an abrupt

drop down to zero is observed. To further investigate the broadening of the Hanle

curve for different initial values of ∆Eos at Bx =0T, we varied the laser power for

the p-shell pumping. The results clearly show an increase of Bcrit as a function of

∆Eos at Bx =0T (Fig. 4.3(c)).


To compare the two sample structures A and B with different electron spin

lifetimes, we first investigated the magnitude of the nuclear spin polarization at

Bx = 0. Testing several p-shell resonances for different QDs did not reveal any

significant difference for the maximal ∆Eos. This result can be explained in two

possible ways. Either the dominant depolarization mechanism is not smaller for the

longer electron spin lifetime or the p-shell pumping is less effective for the longer

electron spin lifetime. Since p-shell pumping relies on a well defined X+ state to

polarize nuclear spins, we contrasted the fluorescence spectrum from sample A with

the one from sample B (Fig. 4.4). For sample B, a very clean spectrum is measured,

where as sample A can obviously be excited to a multitude of states. This is due

to the fact that a longer electron spin lifetime leads to less well defined charge

states [62, 85]. Therefore a smaller nuclear spin polarization rate for sample A is

expected. Hence a longer electron spin lifetime does not result in a larger ∆Eos at

Bx = 0.

According to previous experiments the broadening of the Hanle curve scales with

(a)

(b)

960 962 964 966 968 9700

5000

10000

15000

20000

X+

PLE

count

rate

(cts

/0.5

sec)

Wavelength (nm)

B

952 954 956 958 960 962

0

5000

10000

15000

20000

PLE

count

rate

(cts

/0.5

sec)

X+

Wavelength (nm)

A

Figure 4.4.: Emission at the s-shell.P-shell excitation at saturation for sampleA with a 35nm tunneling barrier (a) and sample B with a 25nm tunneling barrier(b).

∆Eos at Bx = 0 for the same QD on the same sample (Fig. 4.3 (c)). To investigate


-1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

PL c

ircula

r pola

rization

Bx(T)

-1.5 -1.0 -0.5 0.0 0.5 1.0

0

5

10

E

(eV

)

Bx(T)

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

PL c

ircula

r pola

rization

Bx(T)

A B

B

(a) (b)

(c) (d)

-0.4 -0.2 0.0 0.2 0.4

0

5

10

E

OS (eV

)

B

x(T)

A

Figure 4.5.: Hanle measurements for different sample structures. (a-b)Hanle depolarization curves for a QD on sample B with a 25nm tunneling barrierand a QD on sample A with a 35nm tunneling barrier. Arrows indicate field sweepdirection. (c-d) Corresponding energy splitting ∆E between co- and cross- polarizeddetection.

the influence of the electron spin lifetime on Bcrit we performed Hanle measurements

with similar values for ∆Eos at Bx = 0T in sample A and B. In Fig. 4.5 (a-b) typical

Hanle depolarization curves for the two different sample structures A and B are

shown. The most striking difference is the broadening of Hanle curve which is 0.4T

for sample A and 1.5T for sample B. A second difference is the hysteresis, which

reaches up to 0.7T in sample B and vanishes completely for sample A. The third

difference concerns the line shape. All Hanle curves in sample A show a smooth

line shape while for sample B abrupt jumps appear at Bx = Bcrit. This leads to the

conclusion that the nuclear spin polarization at zero magnetic field is not the only

parameter that influences the broadening of the anomalous Hanle curve. In contrast

to what was observed in GaAs droplet dots [84] the magnitude of the nuclear spin

polarization increases during the Hanle measurements and the mechanism which

rotates nuclear spin polarization from the z-axis to the x-axis is unknown. This

makes direct conclusions for the impact of nuclear spin polarization along the

4.5. Resonant measurements on a 35nm tunnel barrier sample 45

z-axis on Bcrit very difficult. Further studies are necessary to fully understand the

increasing nuclear spin polarization along the x-axis in a Hanle measurement.

In addition to the Hanle curve we investigate nuclear spin polarization in z-direction

with the energy splitting ∆Eos between co- and cross- polarized detection of the

emitted light (Fig. 4.5 (c)-(d)). For both samples ∆Eos continuously decreases from

12µeV at Bx = 0 to around zero at 0.4T. While for sample A 0.4T corresponds to

Bcrit, sample B exhibits a broadening of the Hanle curve up to 1.1T. For sample B

the measured ∆Eos = 2µeV in the region between 0.4T and 1.1T is at the resolution

limit for a non-resonant measurement method (see section 3.4.1). Therefore it

is very difficult to claim a persistence of a small nuclear magnetic field along

the z-axis up to Bcrit. In order to achieve a better understanding of the nuclear

spin polarization dynamics in a transverse magnetic field, an increased spectral

resolution is needed.

A valuable alternative to the analysis of the QD fluorescence with a spectrometer

is the resonant excitation of the QD, which was performed with resonance fluores-

cence (RF). A resonant measurement offers an at least four times higher spectral

resolution which is limited due to the optical line width (see chapter 2). In contrast

to non-resonant measurements, RF is an absorption experiment which allows for

a direct access of the optical transitions in the s-shell. With the determination of

the polarization and the energy difference of these transitions, information about

magnitude and direction of nuclear spin polarization is gained.

4.5. Resonant measurements on a 35nm tunnel

barrier sample

To prove that the outcome of the experiment can be reproduced by resonant

spectroscopy, we performed a pump probe measurement with RF on sample A with

a 35nm tunnel barrier. In former measurements a narrow Hanle curve with no

abrupt change in the polarization conservation was obtained. This means that for

the electron spin there is no complete compensation of the external field Bx caused

by the nuclear spin polarization, which can suddenly break down. Therefore we

would also expect a gradual change from the Overhauser splitting to the Zeeman

splitting for increasing Bx.

To study the dynamics of ∆Eos resonantly, a σ+ polarized pump laser is used and

the X+ state was resonantly probed with a weak diagonally polarized laser. At zero

magnetic field, two resonances are observed, which are energetically separated by

the Overhauser shift ∆E=10µeV (Fig. 4.6). With increasing magnetic field Bx a

decrease of ∆Eos is observed. As a result the two resonances merge at Bx = 0.1T.


0 . 1 2 0 . 1 4 0 . 1 60 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

1 T0 . 2 5 T0 . 2 T0 . 1 5 T0 . 1 T0 . 0 5 T0 T

RF co

unt ra

te (ct

s/0.2µ

s)

G a t e V o l t a g e ( V )

Figure 4.6.: RF measurements sample A. Resonant measurements for differentin-plane magnetic fields, performed on a 35nm tunneling barrier sample.

For Bx = 0.2T an evolution into two resolvable resonances is measured because of the

Zeeman splitting. Finally, four resonances can be observed for a large magnetic field

of Bx = 1T. This is due to the fact that in Voigt geometry (transverse magnetic field)

all four transitions of X+ are allowed and linearly polarized [78, 86, 87]. Similar to

non-resonant measurements on the same sample structure, a gradual decrease of the

Overhauser shift can be observed. From the smooth Hanle curve we expect no abrupt

change in the nuclear spin polarization. This is confirmed in resonant measurements.

Hence the resonant pump probe technique allows for investigations of nuclear spin

dynamics in transverse magnetic fields with an enhanced measurement resolution

compared to non-resonant measurements.

4.6. Resonant measurements on a 25nm tunnel

barrier sample

Hanle measurements on sample B with a 25nm tunnel barrier showed an anoma-

lous broadening of the Hanle curve. First, we would like to reproduce these features

which are the abrupt jumps from circular to linear polarization of the emitted

photons, hysteresis and a large value for Bcrit on the order of 1.5T. In a second

step we aim at further investigating the mechanism which rotates the nuclear spin


polarization from the z- to the x-axis.

4.6.1. Gate properties

In order to ensure good charge screening, we rely on the diode structure and in

particular a well grounded back contact. Without the latter, the optically generated

charges get trapped and can result in a reduced frequency bandwidth and thus in

a large RC time constant. It turned out that the back gate of the sample to be

studied was difficult to contact. This is most certainly due to the degradation of the

sample material. Therefore special attention has to be paid to the contact quality.

To test the frequency bandwidth of the gate, we applied a square wave modulation

with a large amplitude such that the two Stark shifted resonances can be resolved

with a spectrometer (Fig. 4.7). By increasing the modulation frequency to 700Hz a

merging to one broad resonance is observed. This is a consequence of the reduced

frequency bandwidth and confirms an incomplete grounding of the back contact.

Therefore the charge screening in the sample structure is restricted.

Not only for a fast gate modulation a good charge screening is needed. Particu-

973.0 973.2 973.4 973.6

1000

2000

3000

counts

/0.5

s

Wavelenght (nm)

2Hz

200Hz

700Hz

Figure 4.7.: Gate modulation at different frequencies. PL counts versus wave-length for different gate modulation frequencies. At 2Hz there are two Stark shifted,well separated resonances observable. For a modulation frequency of 200Hz alreadya merging of the two resonances starts, which evolves at 700Hz to one broad reso-nance. This is due to the incomplete grounding of the back contact.

larly, a strong p-shell pump laser generates a lot of charge carriers in the sample.

Without a well grounded back contact, these charges are trapped and can lead

to fluctuations and a broadening of the resonances in the subsequent measuring

process. In a pump probe experiment performed on sample B, very broad optical

line widths were measured independent of the polarization of the pump laser. As a


consequence the Overhauser splitting could not be resolved.

Even though the sample exhibits a reduced frequency bandwidth, a continuous

wave (cw) RF measurement is still possible because it does not rely on fast charge

screening. A drawback of a cw experiment compared to a pump probe experi-

ment is the increase of the background counts which arises from the incoherent

fluorescence. Whereas in PLE measurements incoherent florescence was spectrally

resolved, it is an additional constant signal for cw RF experiments since the two

fluorescence signals cannot be distinguished. Therefore the signal to noise ratio of

a cw experiment is substantially reduced compared to a pump probe experiment.

4.6.2. Resonance fluorescence background

Performing a RF measurement as in section 4.5 but for a sample with a smaller

tunnel barrier leads to a substantially different outcome. Color coded RF counts

for a decreasing magnetic field Bx versus scanned gate voltage are plotted in

Fig. 4.8 (a). A sudden change of the background together with an increase of the

number of resonances can be observed at Bcrit =1.2T. As discussed earlier, the

main contribution of the RF background arises from the PLE emission. Therefore

the averaged RF background and the emitted σ+ photons in a PLE experiment

versus Bx are shown in Fig. 4.8 (b). In both cases, an abrupt decrease at Bcrit can

be observed. As discussed above, the PLE photons exhibit a strong polarization

preservation i.e. σ+ for Bx < Bcrit. For larger magnetic fields this polarization

preservation breaks down, resulting in a linearly polarized emission (see section 4.4).

As a consequence, the PLE σ+ count rate exhibits a sudden decrease of 75% at Bcrit.

The same step would be observed for a σ+ polarized RF background. However,

the measured decrease in the RF background is only 5%, corresponding to a 7%

circularly polarized RF basis. This is below the measurement uncertainty of the

polarization measurement equipment. Therefore the sudden decrease of the RF

background has a physical origin and is an indication for the breakdown of the

nuclear spin polarization.

Additionally, an increase of the RF background from zero to 0.3T can be seen

in Fig. 4.8 (b). This is also observed for the PLE σ+ emission. Although, the

reason for this increase is unclear, we can rule out any change in the position of

the QD, since the resonant signal does not change its magnitude. Moreover the

increase of the RF background is not a general feature and was only observed for

this particular QD. This is not particularly surprising since the line shape of the

anomalous Hanle curve differs from QD to QD.


-0.63 -0.62 -0.61 -0.60 -0.590.0

0.5

1.0

1.5

2.0

Gate voltage (V)

Bx (

T)

3.4E+04

3.8E+04

4.2E+04

4.5E+04

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

50

100

150

200

250

Bx(T)

PLE

+ c

ount

rate

(cts

/s)

32500

35000

37500

40000

42500 RF

backgro

und (c

ts/s

)

(a)

(b)

Figure 4.8.: CW RF measurements sample B. (a) RF measurements for de-creasing in-plane magnetic field Bx versus gate voltage. White lines are a guide tothe eye indicating the Zeeman splitting. Performed on a 25nm tunneling barriersample (B).(b) Averaged background from RF measurements (blue) as a functionof Bx in comparison with the emitted σ+ photons in a PLE experiment (black). Anabrupt jump at Bcrit is observable in both cases.

4.6.3. Extraction of the resonance positions

Not only the abrupt change in the RF background at Bcrit is an indication for the

breakdown of nuclear spin polarization but also the number of resonances before

and after Bcrit =1.2T. In contrast to PLE measurements, the increased spectral

resolution of the RF measurements allows to measure four resonances above Bcrit

for the first time. Line cuts below and above Bcrit of the measurements shown

in Fig. 4.8 (a) are plotted in Fig. 4.9 (a). Below Bcrit we observe a pair of split

resonances separated by the hole Zeeman splitting (|gh|µBBx). This observation

is fully consistent with the cancellation of Bx with an in-plane Overhauser field.

Above Bcrit the nuclear spin polarization breaks down, resulting in an additional

electron Zeeman splitting (|ge|µBBx) [20]. The expected fourth resonance lies just


-0.63 -0.62 -0.61 -0.60 -0.59

0

2000

4000

6000

8000

10000

ge

BB

x

gh

BB

x

gh

BB

x

RF

counts

(cts

/0.5

s)

Gate Voltage (V)

B=1.05T

B=1.35T

(a)

(b)

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

30

35

40

E

(eV

)

Bx(T)

ge=0.45

gh=0.28

Figure 4.9.: Extracted positions of the resonances (a) Plot of two line cutsfrom the measurements shown in Fig. 4.8(a). Below Bcrit, at B = 1.05T only tworesonances are observed (black), that are split by the hole Zeeman energy ghµBBx.Above Bcrit, at B = 1.35T the nuclear spin polarization breaks down, resultingin the expected electron and hole Zeeman splitting (red). The background is sub-tracted. (b) Extracted energy shift ∆E between different energy transitions of themeasurements shown in Fig. 4.8(a). Arrows indicate the color coded field sweepdirection. Black lines show expected Zeeman splitting; electron and hole g-factorswere determined in an independent measurement.

not within the measurement range.

The energy differences between the various resonances extracted from Fig. 4.8

(a) are plotted as a function of the magnetic field Bx in Fig. 4.8 (b). The data

for an increasing magnetic field is taken from an additional measurement. The

electron and hole Zeeman splitting is indicated with black lines, corresponding

g-factors were determined in independent measurements. The data shows a hys-

teresis, as it is expected from Hanle experiments (Fig. 4.5). At Bcrit the resonances

jump to the energy positions that are calculated from the electron and hole g-factors.

4.6.4. Transition between Overhauser shift and hole Zeeman

splitting

Nuclear spins are polarized along the z-axis at Bx = 0T and Fig. 4.8 implies a

nuclear spin polarization mainly along the x-axis for Bx ∼ Bcrit. To investigate

the evolution of the nuclear spin polarization direction between these two points in


(a)

(b)

-0.60 -0.59 -0.580.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Gate voltage (V)

Bx(T

)

-5 0 5 10 15

0

1000

2000

3000

4000

5000

RF

count

rate

(cts

/sec)

Detuning (eV)

1T

0.45T

0.1T

0T

Figure 4.10.: Transition between Overhauser shift and Zeeman splitting.(a) Resonant measurements for increasing in-plane magnetic fields below Bcrit. Res-onant excitation laser is vertically polarized with respect to the QD axes. (b) Linecuts for selected magnetic fields Bx.

more detail, a resonant RF measurement was performed in an increasing magnetic

field below Bcrit and with a smaller step size of Bx = 0.025T. The result is shown

in Fig. 4.10 (a) after background subtraction. Three different regions can be

distinguished. At Bx = 0T the Overhauser splitting is visible. An intermediate

regime exists at Bx = 0.45T before the hole splitting dominates at Bx = 1T. Re-

spective line cuts that represent the three different regimes are shown in Fig. 4.10(a).

In the first regime, at Bx = 0T the two resonances are not equally strong.

This height difference arises due to depopulation of the ground state caused by

the pump laser. To illustrate the influence of the pump laser on the RF signal

we performed a simulation of the underlying four level system. The calculations

are based on the master equation solved in steady state; incoherent transitions are

included with a Lindblad form [88]. An energy level diagram that indicates also

the considered decay rates is shown in Fig. 4.11 (a). For simplification we did not

consider a dynamic nuclear spin polarization (DNSP) that scales with the pump


-6 -4 -2 0 2 4 60.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

3

3+

4

4)

0.1

50

Detuning ()

pump

:

200

ΔEOS

pump

(b) (a)

WL

g2

g1

Figure 4.11.: Influence of the pump laser. (a)Energy level diagram with theconsidered rates for the simulation shown in (b) (b) Simulated RF spectra for in-creasing pump rate. A strong pump laser leads to an asymmetry of the two peakssplit by the Overhauser shift.

laser power. As a result the two excited states |⇓⇑↓> and |⇑⇓↑> exhibit a constant

splitting ∆Eos as a function the laser power. The influence of the pump laser on

the system is simulated with a rate Γpump from |⇓> to |⇓⇑↓>. Additionally the

spontaneous emission rate Γ from both excited states was set to (2πτ)−1 =250MHz,

where τ is the exciton lifetime. A coupling between the two excited states arises

from co - tunneling and is simulated with γ1 = 0.001Γ. To consider incomplete

hole spin pumping, the two ground states are coupled with γ2 = 0.001Γ. The

resonant linearly polarized laser has a Rabi frequency of ΩL = 0.1Γ, which is

below saturation and drives both transitions, |⇓>→|⇓⇑↓> and |⇑>→|⇑⇓↑>. The

detuning is given as δ in Fig. 4.11 (a). Because the emitted RF photons from

the transitions |⇓⇑↓>→|⇓> and |⇑⇓↑>→|⇑> are measured, we simulate the RF

spectra with the population of the excited states (ρ33 + ρ44) multiplied with the

spontaneous decay rate Γ. The simulated RF spectra for different rates Γpump are

shown in Fig. 4.11 (b). Without considering a pump laser the two peaks, which are

split by ∆Eos, are equally strong. Once the pump laser is considered, an asymmetry

can be observed. In particular, the emitted RF photons from the excited state

|⇑⇓↑> are more pronounced compared to |⇓⇑↓>. This is due to the pump rate

Γpump, which leads via cotunneling γ1 to an increased population of the state |⇑⇓↑>.

Experimental data in Fig. 4.10 (b) show, that the asymmetry is partially cancelled

at Bx = 0.1T. The reason is most likely a mixing of the ground states with a finite

transverse magnetic field which corresponds to an increased rate γ2.

Concerning the intermediate regime at Bx = 0.45T, four resonances would be

expected due to the superposition of the Overhauser shift and the hole Zeeman

splitting. It is clear that the strength of these resonances depends on the optical


(a) (b)

ghμBBx

s- s+

ΔEOS

z

z

z

( + )

( - ) z -10 -5 0 5 10 15 20

0

500

1000

1500

2000

2500

3000

lin

s-

s+

RF

co

un

t ra

te (

cts

/s)

Detuning (eV)

Figure 4.12.: Polarization dependence at Bx=0.45T. (a) Energy diagram ofX+ in a transverse magnetic field Bx < Bcrit for the special case Bn,x = −Bx.(b) Resonant Hanle measurement at Bx = 0.45T for circular and linear resonantexcitation.

selection rules. To gain a better understanding of the polarization of the optical

transitions we once more consider the Hanle measurements. The main property of

anomalous Hanle curves is the PL circular polarization conservation up to Bcrit.

This implies an electron spin polarization along the laser excitation axis (z-axis),

resulting in circularly polarized transitions. This situation is visualized in Fig. 4.12

(b). It shows an energy level diagram for X+, where the excited states are polarized

along the z-axis and separated in energy by ∆Eos. In contrast to the electrons,

the holes are only influenced by the externally applied in plane field Bx. This

leads to the eigenstates |⇓> − |⇑> and |⇓> + |⇑> which are separated by the

hole Zeeman splitting. The polarization of the four possible transitions is given

by the electron spin polarization of the excited state. Therefore, the transitions

|⇓⇑↓>→ (|⇓> − |⇑>) and |⇓⇑↓>→ (|⇑> + |⇓>) should be σ+ polarized while

|⇑⇓↑>→ (|⇓> − |⇑>) and |⇑⇓↑>→ (|⇑> + |⇓>) should be σ− polarized. As a

consequence a resonant measurement with a linearly polarized laser results in four

resonances which are separated by the hole Zeeman splitting and ∆Eos. These two

energy shifts are most likely very similar in the measurement shown in Fig. 4.10

(b) at Bx = 0.45T. As a result, one large resonance arises from the two diagonal

transitions and is flanked by two smaller ones.

To further confirm the polarization of the involved optical transitions, we per-

formed a resonant measurement with circularly polarized light at Bx = 0.45T.

RF spectra for a linear, σ+ and σ− polarized laser excitation are shown in Fig. 4.12

(a) after subtraction of the background. While three resonances are observed for the

linear RF spectra, only two resonances are measured for both circularly polarized

excitations. The two resonances are separated by the hole Zeeman splitting. Due


to the strong variations in the background, we measured only a short detuning

range of the laser. With additional measurements it was verified that there exist no

additional resonances for σ+ and σ− excitation. Moreover, the resonance positions

between the different polarizations of the resonant laser cannot be compared.

During the change of the RF polarization basis it is very likely to trap charges in

the sample which lead to an energy shift. The signal to noise ratio differs for various

polarizations which can be explained as follows. For a resonant σ− excitation, σ+

polarized RF photons are detected and a large background is measured due to the

mainly σ+ polarized PLE emission. Therefore the signal to noise ratio for the σ−

RF spectrum is worse than for the σ+ RF spectrum in Fig. 4.12 (a).

One hast to keep in mind that for RF only the photons that have a polarization

component orthogonal to the resonant laser polarization can be detected. According

to the transitions plotted in the level diagram of Fig. 4.12 (b) no signal should be

measured for σ+ or σ− excitation. This situation changes if there is an interaction

between the two excited states |⇓⇑↓> and |⇑⇓↑>, such that an excitation with σ+

and σ− can be followed by a decay of σ− and σ+ polarized light, respectively. Such

an interaction could arise due to a net field along the x-direction. Additionally,

PLE experiments show a polarization conservation of only 60% for Bx = 0.45T.

Therefore, an electron spin flip has to take place either in the p- or in the s-shell of

the QD, which further supports the hypothesis of an interaction between the two

excited states |⇓⇑↓> and |⇑⇓↑>.

4.7. Discussion 55

4.7. Discussion

By resonantly measuring nuclear spin polarization with RF in transverse magnetic

fields we achieved on the one hand a higher spectral resolution and on the other hand

a direct measurement of the X+ transitions. Particularly, the polarization of the

X+ transitions as well as the population of the ground states can be determined.

Due to the fact that only two resonances are obtained for circular RF spectroscopy

at Bx = 0.45T and because they exhibit a splitting of exactly |gh|µBBx, we conclude

that the electron in the excited state defines the polarization of the optical transi-

tions. Hence the transitions |⇓⇑↓>→ (|⇓> − |⇑>) and |⇓⇑↓>→ (|⇑> + |⇓>) are

σ+ polarized while |⇑⇓↑>→ (|⇓> − |⇑>) and |⇑⇓↑>→ (|⇑> + |⇓>) are σ− polar-

ized. This conclusion is also supported by the linear RF spectrum which shows that

all four transitions contribute to the signal. The observation of circularly polarized

optical transitions are fully consistent with a cancellation of the external field by

Bxeff, that in turn hinders the electron spin rotation.

To allow for a better understanding of the evolution of the nuclear spin polarization

direction in a transverse magnetic field, Bnucz and Bnuc

x were extracted from the mea-

surements shown in Fig. 4.8 (a) and plotted vs Bx (Fig. 4.13). This extraction is

only possible under the assumption that Bnucx cancels completely the external mag-

netic field Bx for circularly polarized transitions in X+. Starting from Bnucz =0.67T

at Bx = 0T, the nuclear field is gradually decreasing to 0 at Bx = 0.5T. This is

far below Bx = Bcrit. On the contrary, the magnitude of the nuclear field along the

x-axis is increasing. A complete cancellation of the applied magnetic field Bx can be

observed up to Bcrit = 1.7T, where the nuclear spin polarization breaks down and

where the Zeeman splitting that is expected for an experiment in Voigt geometry

is restored. The magnitude of the nuclear field at Bcrit along the x-axis is roughly

2.5 times larger than the nuclear field along the z-axis for Bx = 0T. This means

that the nuclear field does not only rotate into the xy-plane but also increase with

increasing in-plane magnetic field.

Sallen et al. [84] explained the origin of the transverse nuclear spin polarization in

Hanle experiments on droplet QDs with the Knight field. The magnetic field expe-

rienced by the nuclear spins is given by a superposition of the external transverse

magnetic field and the Knight field. The total magnetic field is therefore slightly

tilted towards the z-axis. Thus, the nuclear spin polarization has a contribution

along the x- and the z-axis. This model was successfully applied to describe the

line shape of Hanle curves in droplet QD [84]. For the case of droplet QDs the

broadening of the Hanle curve is in the order of a few 10mT, which is exactly the

range of the Knight field.

These broadening effects occur for one order of magnitude smaller values of Bx

than the anomalous Hanle effect repeated in this work for InGaAs QDs. Compared


0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

Bx(T)

Bn

uc

z (

T)

0.0

0.5

1.0

1.5

Bn

uc

x (T

)

Figure 4.13.: Nuclear field changes in Bx. Extracted nuclear field magnitudeand direction versus transverse magnetic field Bx from Fig. 4.10.

to droplet GaAs QDs, self assembledInGaAs QDs are heavily strained which leads

with the associated nuclear quadrupolar interactions (QI) to a sizable Bnucx [40, 89].

This strain is an inherent property of the self-assembled growth process which relies

on the lattice mismatch between GaAs and InAs. As a result the QD strain gra-

dient coincides predominantly with the QD growth direction (z-axis) [90]. Hence

the QD experiences an electric field gradient along the z-axis. Due to the non

spherical charge distribution of nuclear spins, they couple to the electric field gra-

dient via quadrupolar moments (see chapter 3). The corresponding level separation

leads with the relation gNµNBQ = ~ωQ to an effective magnetic field BQ of sev-

eral 100mT [51, 91]. Particularly, in a small transverse magnetic field, the nuclear

spins do not feel the presence of Bx to first order. Therefore Bnuc is always oriented

along the z-axis. Such a static magnetic field along the growth direction can lead

to a nuclear spin polarization along the x-axis as follows. The total magnetic field

experienced by the electron is given by Btot = Bnuc + Bx. Quadrupolar interac-

tions stabilize the nuclear spin polarization along the z-axis, resulting in a tilted

Btot. Therefore the electron spin precesses around this tilted magnetic field. Via

hyperfine interaction the electron spin polarization is transferred to nuclear spins,

resulting in a nuclear spin polarization in the xz-plane. In addition, the polariza-

tion along the x-axis is further stabilized by Bx. Furthermore, the resulting Bnucx

compensates Bx and reduces the total electron spin splitting which leads to a high

dynamic nuclear spin polarization (DNSP) rate.

However, this model cannot be confirmed with the data shown in Fig. 4.13 because

Bnucz is below the measurement resolution of 0.1T for Bx > 1T. Probably the model

is too simple to fully describe the influence of QI on the Hanle effect. The assump-

tion of a strain gradient along the z-direction is only valid for a spatial average over

4.7. Discussion 57

the QD. Most nuclei do not experience a strain gradient along the z-direction, which

leads to a different behavior of the nuclear spin system in a transverse magnetic field.

Not only the direction but also the strength of strain can vary within the QD. An

exact description of various inhomogeneities might lead to a better understanding

of the experimental data.

On the other hand, the abrupt jump of nuclear spin polarization along the x- axis

together with electron spin polarization to zero in combination with a hysteresis

could indicate an alternative view on the anomalous Hanle effect, namely the oc-

currence of a spontaneous polarization and phase transition. However, these results

are quantitatively different from the theoretically predicted quantum phase transi-

tions [80, 83], since an abrupt decrease of the nuclear spin polarization along the

z-axis at Bcrit has not been observed. Even though there are a lot of similarities,

the experimental system deviates in several points from the theoretical model. Es-

pecially, quadrupolar interactions and the inhomogeneous hyperfine interaction are

strongly influencing nuclear spin dynamics and might lead to a different behavior

than what is theoretically predicted. To rule out the influence of the QI, a study of

strain free QDs should be considered that exhibit the same degree of electron spin

polarization as in InGaAs QD which is around 80%. For GaAs droplet QDs only

40% electron spin polarization was achieved. This value might be improved with

a quasi-resonant pumping [84]. It would be interesting to investigate Hanle mea-

surements on GaAs droplet QDs with a the same degree electron spin polarization

as in InGaAs QD, to see if the nuclear spin evolution is closer to the theoretical

predictions of phase transitions.

Coherence of Optically Generated Hole Spins 59

5. Coherence of Optically Generated

Hole Spins

To realize a quantum bit a single hole spin in a semiconductor quantum dot

has potential advantages over an electron spin. One major advantage of

holes is that they have a strongly reduced hyperfine interaction with nuclear

spins, which leads to less dephasing. We measured the coherence time T ∗2and T2 of an optically generated hole inside an InAs/GaAs QD embedded

in an n-doped diode structure in time domain.

5.1. Introduction

Due to their relatively long lifetimes, semiconductor qubits are promising can-

didates for quantum information processing. Further interesting properties are

their potential for integration on a chip [14, 92, 93] and ultrafast coherent optical

control [12, 61, 94, 95]. In particular, electron spins confined in self-assembled

InAs/GaAs QDs have been extensively studied as semiconductor qubits over the

past years. However, such electron spin qubits strongly couple to the nuclear

spin reservoir via hyperfine interaction (see chapter 4). The variations in the

effective magnetic field that result from the fluctuating nuclear spins, induce

electron spin precessions at various frequencies. As a consequence, reduced

coherence times of T ∗2 = 1 − 2ns have been measured [61]. One method to

work around the adverse effect of the slowly fluctuating effective magnetic field

is the spin echo experiment. In such an experiment the phases of the spins are

refocused. As a result, an increased dephasing time of T2 = 3µs can be achieved [61].

A completely different approach to realize a longer coherence time is to work

with a hole instead of an electron. In contrast to an electron, the coupling to

nuclear spins is at least one order of magnitude weaker for a hole [19, 44, 96–98].

Therefore, longer coherence times are generally observed. Two different methods

were used in recent experiments to charge the QD with a single hole. While

hole δ-doped samples are stochastically charged, embedding the QD layer into a

60 Coherence of Optically Generated Hole Spins

p-doped diode structure allows for a deterministic charging of the QD with holes.

Both sample structures were studied by De Greve et al. [99], and they observed

similar values for of T ∗2 = 2.3ns for both samples. Several measurements were

performed on QDs embedded in a p-doped structure, exhibiting hole coherence

times of T ∗2 =20-100ns [11, 100, 101]. Is the latter method combined with a spin

echo measurement, a coherence time of T2 = 1.1µs can be obtained [99].

One common disadvantage of p-doped samples is the broad X+ line width of

∼ 7µeV as compared to the 2-3µeV of n-doped structures. At the ultimate limit of

lifetime broadening, the coherence time is equal to the radiative lifetime τ = 1ns.

This would correspond to an exciton line width of Γ = ~/τ ∼ 1µeV for the

investigated QDs. For high quality n-doped samples, usually optical line widths

of twice this value are measured [15, 31, 65]. While this broadening can either be

explained by enhanced charge fluctuations in the vicinity of the QD or nuclear spin

fluctuations [26, 65], the lager value for the X+ line width in p-doped samples is

very likely due to charge fluctuations. With the lower mobility of p- compared

to n-doped samples, the charge screening is less distinct leading to more charge

fluctuations. According to De Greve et al. [99] this charge noise contributes to

inhomogeneous dephasing which limits the coherence time T ∗2 . Therefore prolonged

coherence times should result for a method that combines the advantage of holes

together with the reduced charge noise in n-doped samples. This is the focus of the

presented work in this chapter.

The question still remains how to most accurately measure the coherence times.

One possibility to determine T ∗2 is coherent population trapping (CPT) [102, 103].

The width and depth of the dip at the two photon resonance of a Λ system

determines the coherence of its ground states. CPT was used for determining

coherence properties of holes in QD with p-doped back contacts [11] and singlet

triplet states in coupled QDs [104]. The main advantage of CPT is that it is based

on spectroscopy and that there is no accurate control over optical pulses needed.

Disadvantages are the limited accuracy of determining the depth of the CPT dip

which is given by the signal to noise ratio [11] and that the method is restricted to

determining T ∗2 times. Additionally, CPT on electron spin states is superimposed

by dragging [40] which makes a distinction between nuclear spin effects and electron

spin coherence properties very difficult.

Latest experiments are based on a more versatile tool to measure T ∗2 and T2 in the

time domain. Press et al. [12] were the first to achieve a complete coherent control

over an initialized electron spin state in a quantum dot with picosecond optical

pulses. Subsequently, de Greve et al. [99] demonstrated a successful implementation

of the same measuring method for a single hole qubit on a p- doped sample.

5.2. Spectroscopy on X+ 61

In this chapter we present results from coherence time measurements that are

performed for the first time on optically generated holes in n-doped samples. In

analogy to Press et al. [12] we measure T ∗2 and T2 in the time domain.

5.2. Spectroscopy on X+

The sample structure consists of a single layer of self-assembled InGaAs QDs,

which is embedded into an n-doped diode structure. This allows for a deterministic

charging of the QD with single electrons. To perform optical spectroscopy on a

positively charged QD, an additional laser is used to generate a positive charge.

Further details on generating the positively charged hole by optical means are given

in Chapter 2 of this thesis.

A typical resonance fluorescence (RF) spectrum of X+ as a function of the energy

- 1 5 - 1 0 - 5 0 5 1 0 1 5

5 0 0 0

1 0 0 0 0

1 5 0 0 0

RF co

unt ra

te (ct

s/s)

D e t u n i n g ( µe V )

2 . 3 µe V

X +

Figure 5.1.: Optical line width of X+ RF signal of X+ as a function of theenergy detuning. The black line is a Lorentzian fit to the data.

detuning is shown in Fig. 5.1. From the Lorentzian fit we can extract an optical line

width of 2.3µeV. Such a narrow optical line width is characteristic for an InGaAs

QD embedded into an n-doped structure, similar values have been measured for

the X− transition in the same QD. In comparison, p-doped samples exhibit with

7µeV [101] much broader optical line width on X+. This broadening of the optical

line width is most likely given by enhanced charge fluctuations in p-doped samples

compared to n-doped structures, as lined out in the previous section. This charge

noise contributes to inhomogeneous dephasing which limits the coherence time T ∗2 .

Therefore prolonged coherence times should result for optically generated holes in

n-doped samples.


5.3. Initialization and Readout

Reliable initialization and readout mechanisms are a prerequisite for a coherent

manipulation of a single hole spin. In the first part of this section the initialization

of the hole qubit into the spin up state is discussed. For this purpose, hole spin

pumping is used [15, 78]. The initialization mechanism is schematically shown in

Fig. 5.2 (b). First, we apply a finite transverse magnetic field (Voigt geometry).

This leads to a Zeeman splitting of the hole spin states. In Voigt geometry, the

diagonal transitions |⇓⇑↓>→|⇑> and |⇑⇓↑>→|⇑> are allowed. After exciting

the |⇓>→|⇓⇑↓> transition with a laser, the decay takes place via the diagonal

transition to |⇑> or back to the ground state |⇓> with equal probabilities. Since the

pumping rate to |⇑> is larger than the decay rate of the hole spin, an initialization

into the hole spin up state |⇑> can be achieved leading to a reduced absorption

of the laser and here in a reduced RF signal. The sample used in this work

exhibits incomplete spin pumping. This is why there is still 30% of the signal left.

Experimental evidence that the diagonal transitions are allowed is given with a

single laser RF measurement. Corresponding data is shown in Fig. 5.2 (a). Four

transitions are visible in the color coded plot of RF count rate as a function of gate

voltage and laser wavelength.

In contrast to X−, spin pumping for holes is possible without any external

-1750 -1500 -1250 -1000 -750 -500

970.62

970.60

970.58

970.56

970.54

970.52

970.50

Gate Voltage (mV)

Wavele

ngth

(nm

)

(b)

ghμBBx

WL

geμBBx

(a)

A

Figure 5.2.: X+ in Voigt geometry. (a) RF measurement of X+ versus gatevoltage and laser wavelength in an in-plane field of 4T (Voigt geometry). At thegate voltage indicated with A, Ramsey measurements are performed. The resultsare presented in section 5.7. (b) Energy level diagram for hole spin pumping. Afterresonantly exciting |⇓>→|⇓⇑↓> , the allowed diagonal transition in Voigt geometry,|⇓⇑↓>→|⇑>, leads to hole spin pumping.

magnetic field. This is due to the limited hyperfine interaction of holes with nuclear

spins resulting in a larger pump rate compared to the hole spin relaxation time [78].

Due to the fact that diagonal transitions are allowed in Voigt geometry, hole

5.4. Rabi Oscillations 63

spin pumping is even more efficient for finite magnetic fields and shows the same

magnetic field dependence as for electron spin pumping [15, 105]. An advantage of

this simple initialization process in Voigt geometry is the fact that it can be used as

well for the readout. If the laser is resonantly applied to the |⇓>→|⇓⇑↓> transition,

the RF signal strength is proportional to the occupation of the ground state |⇓>.

The coherent oscillation between the two hole spin states |⇓> and |⇑> is measured

as a projection onto the z-axis. The respective oscillation of the occupation in the

ground state |⇓> is therefore directly measured via the RF signal strength.

5.4. Rabi Oscillations

(b) (a) x basis

z basis

z

z

z

z

Wrot

wx

Wrot

ħwx

Figure 5.3.: Scheme for Rabi oscillations. (a) Level diagram in x basis forfinite magnetic field Bx. A slightly detuned (210MHz) circularly polarized rotationpulse is driving all four transitions, which are linearly polarized in Voigt geometry.(b) Level diagram in z basis for finite magnetic field Bx. Bx results into a Larmorprecession of the hole spins around the x-axis (red arrow).

After initializing the hole spin state by optical pumping, a spin rotation (Rabi

oscillation) is performed with a 4ps long, σ+ polarized laser pulse. To avoid a

population of the excited state, the laser is additionally detuned by ∆ = 210GHz.

The effect of the laser pulse can be easily understood with the help of the schematic

drawing in Fig. 5.3. In the x-basis the hole spin in the ground state is either

parallel or antiparallel to the applied magnetic field Bx. All optical transitions are

linearly polarized due to the optical selection rules in Voigt geometry. Because

we use a circularly polarized laser pulse, all four transitions are driven (Fig. 5.3

(a)). Now we consider the same situation in the z-basis (Fig. 5.3 (b)), where the

hole spin orientation is along the growth direction. In this case the two allowed

optical transitions are circularly polarized, which means that Ωrot drives only one


optical transition. Additionally, the Zeeman splitting due to Bx results in coherent

mixing of the ground and excited states. After the initialization of the hole spin in

|⇑>x= eiφ/2 |⇑>z +e−iφ/2 |⇓>z, the laser pulse induces an ac-Stark shift of |⇑>z

relative to |⇓>z. Thereby, a relative phase shift φ occurs, that is proportional to

the laser pulse duration and the laser power Prot. In other words, a laser pulse is

equivalent to an effective magnetic field along the z-axis which induces coherent

oscillations between |⇑> and |⇓>. (If not indicated with an index, |⇑> and |⇓>are in the x-basis.)

Due to the fact that every hole spin state can be represented by a superposition

of the states |⇓> and |⇑> and their relative phase, spin manipulations are best

visualized on a unit sphere called Bloch sphere. A schematic drawing of the Rabi

oscillations on the Bloch sphere is shown in Fig. 5.4 (a). After the initialization in

the |⇑> state, a rotation of the hole spin around the z-axis by an arbitrary angle

can be achieved if the rotation pulse power Prot is chosen correspondingly.

To determine the laser pulse power corresponding to a π pulse, the emitted

0.0 0.5 1.0 1.5 2.0 2.5

500

1000

1500

π

RF c

ount

rate

(cts

/0.1

s)

Prot/Pπ

a) b)

y z

x

Figure 5.4.: Rabi oscillations. (a) Visualization of Rabi oscillations on the Blochsphere. The dark red arrow indicates the initialization in the |⇑> state. With thelaser power, the rotation angle between |⇑> and |⇓> can be controlled (light redarrows). (b) Rabi oscillations between the hole spin states are observed in theoscillating RF signal as the rotation pulse power is increased.

RF photons are measured according to the readout scheme as described above.

Fig. 5.4 (b) shows the oscillating RF count rate as a function of the normalized

laser power Prot. As a consequence of the applied readout scheme, the maximal

number of photons indicates the maximal occupation of the |⇓> state. Because the

experiment is initialized in the |⇑> state, the laser power at the maximal count

rate corresponds to a π rotation on the Bloch sphere. As a result of the previously

mentioned imperfect hole spin pumping, the visibility of the Rabi oscillation is

5.5. Ramsey measurements 65

reduced. This explains that the minimal observed photon number is 250cts/0.1s

above the dark count rate of the APD. A reduced visibility, however, has no

influence on the accuracy of the determination of Pπ. The accurate control over the

rotation angle between |⇑> and |⇓> allows to conduct Ramsey experiments and

hence the determination of the coherence time T ∗2 .

5.5. Ramsey measurements

a) b) c)

y z

xBx

Figure 5.5.: Ramsey scheme. (a) After initialization in the |⇑> state, a π/2pulse is applied. (b) The in plane field leads to a rotation of the hole spin aroundthe vertical axis at the Larmor frequency. (c) The rotation is stopped after a delaytime τ with a second π/2 pulse. Depending on the phase evolution, the π/2 pulsebrings the spin back to its initial state |⇑> or to the opposite spin state |⇓>.

With the Rabi experiments in the previous section, we demonstrated that a rota-

tion of the hole spin by an arbitrary angle around the z-axis is feasible. To have a

full control over the Bloch sphere, a rotation around a second axis is needed. with

the in plane magnetic field causing a Larmor precession of the hole spin, a rotation

around the x-axis is established. This effect is exploited for Ramsey interferometry.

In a first step after initialization, a π/2 pulse is applied, that brings the hole spin

to the equator of the Bloch sphere. During the subsequent delay time τ the spin

rotates around the x-axis at the Larmor frequency. The phase evolution determines

the probability of bringing the spin back to either the initial state or to the opposite

spin state with a second π/2 pulse applied after the time τ (Fig. 5.5). Finally, the

measurement of the spin population as a function of the delay time τ leads to a

sinusoidal oscillation of the RF counts at the Larmor frequency (Ramsey fringes).

The effective magnetic field causing the Larmor precession varies slightly in time

and so does the precession frequency. Therefore, averaging that is needed to obtain

a reasonable signal strength leads to an exponential decay of the Ramsey fringes

with a time constant T ∗2 .

To experimentally achieve two π/2 pulses separated by a delay time τ , the laser


Translation stage

B

A

4 ps

4 ps

To sample

Pulsed laser

Figure 5.6.: Delay time control.

beam is split into two optical paths A and B (see Fig. 5.6). Practically, there are

two different possibilities to control the delay time τ . The first one comprises a

constant prolongation of the optical path A with an optical fiber. The second one

delays the optical path A continuously by means of a translation stage. Discrete

steps of the delay time represent measurements performed with inserted fibers of

different lengths. Continuous change of τ over an interval of 2.5ns is realized with a

translation stage.

At a gate voltage of -1.113V corresponding to point A in Fig. 5.2 (a) a decay of the

oscillation amplitude can be observed at τ = 20ns, which is followed by a revival

of the signal at τ = 35ns. Ramsey fringes for Vg = −1.13V before and after the

signal has recovered are shown in Fig. 5.7 (b). In both cases a sawtooth shape of the

fringes can be observed. For a further investigation of the strong amplitude modula-

tion of the Ramsey fringes, measurements have been performed at Bx = 4T and for

delay times up to τ = 400ns. To extract the amplitude of the Ramsey oscillations,

a Fourier transformation is carried out. The resulting fringe amplitudes at the Lar-

mor frequency are plotted as a function of the delay times in Fig. 5.8. Data with

similar amplitude modulations for the Ramsey fringes have been observed by Carter

et al. [101]. They attribute this modulation to nuclear spin interactions and propose

a theoretical model that predicts an envelope modulation for the Ramsey fringes of

twice the 115In precession frequency of 75MHz. They argue that even though there

are several other isotopes present leading to nuclear spin effects, 115In is dominant

due to the large nuclear spin of 9/2 and its high abundance. We applied this model to

fit the data in Fig. 5.8 with an exponentially decaying cosine function. An amplitude

modulation frequency of 74.6MHz and a decay time of T ∗2 = 240ns are extracted.

5.5. Ramsey measurements 67

5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

Vg=-1130mV

RF

count

rate

(cts

/0.5

s)

Delay (ns)

34.5 35.0 35.5 36.00

1000

2000

3000

4000

5000

6000

RF

count

rate

(cts

/0.5

s)

Delay (ns)

4.0 4.5 5.0 5.50

1000

2000

3000

4000

5000

6000

RF

count

rate

(cts

/0.5

s)

Delay (ns)

(a)

(b)

p/2 p/2

t

Figure 5.7.: Ramsey fringes. (a) Ramsey fringes measured for two different tran-sitions in X+, shown in Fig. 5.2 (a) . A decay and revival of the oscillation amplitudecan be observed. (b) Magnification of the of a 1.5ns delay window highlighted witha red box in (a), to display the Ramsey fringe shape. After the revival of the oscilla-tion amplitude, at τ = 36ns, oscillations can still be observed with almost unchangedvisibility.

The modulation frequency matches within the measurement uncertainty twice the

expected 115In precession frequency of 75MHz. Not only the amplitude modulation

but also the sawtooth pattern of the Ramsey fringes can be explained with nuclear

spin polarization as follows [101]. Because a different polarization is achieved for

each delay time, the Overhauser shift changes the optical transition relative to the

laser. Therefore, the initialization and measurement process is disturbed, resulting

in the observed sawtooth pattern of the Ramsey fringes. Similar asymmetric pat-

terns were observed in Ramsey experiments with electron spins. [106, 107].


5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 005 0

1 0 01 5 02 0 02 5 03 0 03 5 04 0 0

fringe

ampli

tude (

cts/0.

5s)

D e l a y ( n s )

T 2 * = 2 4 0 n s

Figure 5.8.: T ∗2 measurement. Ramsey fringe amplitude versus delay time witha decaying cosine fit at Bx = 4T. The extracted time constant of the exponentialdecay is T ∗2 =240ns.

5.6. Spin Echo measurements

y z

x

a) b) c)

Bx

Figure 5.9.: Spin echo scheme. (a) After initialization into the |⇑> state, a π/2pulse is applied. (b) The effective magnetic field varies for consecutive measure-ments. Some spins precess slower around the vertical axis because of the smaller fieldstrength (green arrow) and some spins precess faster due to a larger field strength(blue arrow). By applying a π pulse after a time delay T , the slower spins are nowahead of the faster precessing spins. (c) Complete refocusing takes place after atime delay T . Depending on the phase evolution, the π/2 pulse brings the spin backto its initial state or to the opposite spin state.

As it was discussed earlier, T ∗2 can be limited due to inhomogeneous dephasing

that arises from slow fluctuations in the nuclear spin polarization as well as from

charge fluctuations. To circumvent such slow sources of decoherence, spin echo

measurements are carried out.

The advantage of the spin echo measurement over Ramsey experiments is explained

by means of the schematic illustrations in Fig. 5.9. After initialization of the hole

5.6. Spin Echo measurements 69

T+t T-t

p/2 p p/2

4.6 4.8 5.0 5.2 5.4 5.6400

450

500

550

600

650400

450

500

550

600

650400

450

500

550

600

650

4.6 4.8 5.0 5.2 5.4 5.6

T=1467ns

T=204ns

Time offset t (ns)

RF

co

unt

rate

(cts

/0.5

s)

T=630ns

Figure 5.10.: Spin echo measurements. Spin-echo signal as the time offset τ isvaried, for a time delay of 2T = 1467ns, 630ns, 240ns from top to bottom. Magneticfield Bx=4 T.

spin into |⇑>, a π/2 pulse is applied. As lined out before, the effective magnetic

field and therefore the precession frequency around the x-axis are slightly different

for consecutive measurements. This leads to different precession frequencies in the

horizontal plane of the Bloch sphere, as indicated with colored arrows in Fig. 5.9

(b). To rephase the system, an additional π pulse is applied after a delay time T .

Similar to the Ramsey measurements, a second π/2 pulse is then applied and the

spin population is measured as a function of the time offset τ [60, 61]. This measure

to rephase the spins in different realizations of the experiment is effective, as long

as the fluctuations of Beff are slow compared to T .

The spin-echo signal for three different time delays T is shown as a function of

the time offset τ in Fig. 5.10. Between a time delay of T = 204ns and T = 630ns,

only very little decay in the amplitude is observed. For the longest time delay at

T = 1467ns only 10% of the signal amplitude remains. Even though there are not

enough data points for fitting an exponential decay to the signal amplitudes, the


decay time T2 can be estimated to be on the order of 1µs.

5.7. Discussion

We demonstrated experimentally that ultrafast coherent control techniques work

as well for optically generated hole spins as for electron spins in n-doped samples.

From Ramsey measurements we extracted a T ∗2 time of 240ns for optically generated

hole spins. This is a 1-2 order of magnitude larger value than what was measured

with the same method for hole spins in p-doped samples (T ∗2 = 2.3−19ns) [99, 101].

As discussed in the introduction the T ∗2 time is the longer, the less pronounced the

effects of inhomogeneous dephasing are. In other words, n-doped samples exhibit

either less charge noise or weaker fluctuations in the nuclear spin polarization. A

comparison of the optical line widths in X+ reveals for n-doped samples 2-3µeV

which is a much smaller value than the measured 7µeV for p-doped structures.

This large difference can only be caused by charge fluctuations. For that reason

smaller charge induced spectral diffusion does most likely lead to a longer T ∗2coherence time in n-doped samples compared to p-doped structures. To summarize

our results, the method of optically generating holes in QDs brings clear advantages

over conventional techniques.

By contrast, the measured spin echo time T2 for optically generated holes (T2 = 1µs)

and p-doped samples (T2 = 1.1µs) are comparable. Most interestingly also the T2

times for electrons are with 3µs in the same range. Based on dipole dipole interac-

tions of nuclear spins Witzel et al. [108] and Yao et al. [109] predict T2 ∼ 1− 10µs

for a single electron in a QD. In self-assembled QDs this nuclear spin diffusion is

negligible because of the quadrupolar interactions. Recent experiments revealed

that electron mediated nuclear spin diffusion and co - tunneling rather limit the

T2 time [79]. The measured values for T2 are rather surprising, since the weaker

interaction of hole spins with nuclear spins compared to electrons should result

in a longer T2 value. Hence, the limiting factor for the spin echo time of holes

cannot be nuclear spin effects. For p-doped samples, it is very likely that charge

induced spectral diffusion, which already limits T ∗2 , sets an upper boundary for

T2 as well. Regarding charge noise, optically generated holes in n-doped samples

are less affected which should result, compared to p-doped structures, in a longer

spin echo time. Because such a behavior is not observed, it is very probable that

the hole generating process with a weak off-resonant laser is limiting the coherence

properties. To test this hypothesis, hole spin lifetime measurements should be

carried out as a next step.

As expected from the weaker hyperfine interaction of holes with nuclear spins

compared to electrons in InGaAs QDs [19, 44, 96–98], the measured decoherence

5.7. Discussion 71

time T ∗2 = 240ns is longer than for electrons (T ∗2 = 1 − 2ns) [61]. The hole spin

decay is dependent on the orientation of the external magnetic field. This arises

from the simple Ising form∑

k AhkI

zksz that couples the hole to the nuclear spin

system, where Ahk is the coupling of the kth nucleus and Izk and sz denote the z

component of the kth nuclear spin and the hole spin, respectively. In the case of a

transverse magnetic field Bx the nuclear spin fluctuations are perpendicular to the

applied field Bx, resulting in a slow power-law hole spin decay which is estimated to

be on the order of µs. In comparison for zero field or an external field Bz the decay

is Gaussian, similar to electrons, with timescales of tens of nanoseconds [19]. Hence,

the measured two orders of magnitude longer T ∗2 for holes compared to electrons

can be explained with the weak coupling of holes to the nuclear spin system in

transverse magnetic fields.

Even though the interaction between holes and nuclear spins is weak, a strong

amplitude modulation of the Ramsey interference fringes is observed. This is a

result of nuclear spin polarization. It changes the optical transition relative to the

laser via the Overhauser shift leading to an amplitude modulation and a sawtooth

shape of the Ramsey interference fringes [101].

In addition to the results we presented above, our work gives rise to a num-

ber of further investigations. For instance, the fact that n-doped samples allow

to charge the same QD either with an electron or with a hole could be exploited

to directly compare the coherence behavior of both species. This might give new

insights into the interaction of electrons and holes with nuclear spins.

For several interesting experiments it is crucial to achieve a long coherence time. In

particular we have shown that optically generated holes in QDs are promising to

further prolong the spin echo time T2. With their work on quantum teleportation

between a photon and a single electron spin qubit, Gao et al. [107] show in which

direction research might head. Their experiments in combination with longer

coherence time could lead to a realization of spin spin entanglement.

Appendix A

A. Appendix

A.1. Experimental Setup

With a confocal microscope [110] resonant spectroscopy was performed. On the

one hand this allows for a small spot size of ∼ 1µm which is crucial for single QD

spectroscopy, on the other hand a large amount of light emitted from the QD can

be collected with a large numerical aperture (NA) of the objective. The numerical

aperture NA ∝ sinθ ∼ θ is proportional to the half angle θ of the light cone from

a lens that is illuminated with a plane wave. Therefore a higher NA increases the

collection efficiency. We performed all experiments with the lens L1 (see Fig. A.1)

that exhibits an NA of 0.68. Additionally, we used a solid immersion lens (SIL) to

increase the collection efficiency of the emitted light from the sample surface [111].

A four times increased collection efficiency was experimentally observed by compar-

ing the same sample with and without SIL.

To position the QD into the focal spot, the sample is mounted on piezoelectric

xyz-positioners. The described microscope together with the positioners and the

sample are placed inside a bath cryostat which reaches temperatures of 1.4K. Such

low temperatures are necessary to prevent temperature broadening of the QD reso-

nances. Magnetic fields up to 12T can be applied with a superconducting split coil

magnet. This geometry allows for an optical access of the sample along two different

axes. Therefore upon rotation of the sample, a magnetic field transverse (shown in

Fig. A.1) or along the growth direction can be applied.

For PL measurements the QD is excited with diode laser at 780nm, the emitted

photons from the QD are dispersed with a spectrometer (SP) of 75cm focal length

and finally analyzed with a liquid nitrogen cooled, back-illuminated CCD array.

To perform resonant spectroscopy or PLE, two tunable diode lasers (Toptica DL

pro) were available with a tuning range from 915nm to 990nm. The required in-

tensity stabilization for resonant measurement is realized with a double-pass AOM

setup [112], before the lasers are coupled into a fiber. While the dT signal is directly

measured with the photodiode PD1, the dR signal is first coupled into a single mode

fiber (CF) before it is measured on a photodiode. The additional single mode fiber

allows for a mode filtering, which is especially needed in RF measurements. In both

cases, dR/R and dT/T, after passing a preamplifier, the signal is sent to a Lock-

B Appendix

In amplifier. The Lock-In amplifier is synchronized with a function generator that

produces a square wave modulation with a DC offset which is applied to the gate of

the sample. Because of the Stark-shift, the QD transition is modulated in and out

of resonance. This technique is called Stark-shift modulation [31, 71] and allows to

suppress noise on all frequencies except the modulation frequency which is usually

at 777Hz.

Polarization control is needed to perform resonance fluorescence (RF) measurements.

By choosing the polarizer P2 cross-polarized to the polarizer P4 in front of the sin-

gle mode collection fiber (CF) an extinction ratio of ∼ 106 can be achieved. For

RF measurements it is very difficult to optimize the QD position by monitoring

the signal, because every change of the sample position leads to an increase of the

background. To align the setup and to monitor the QD position a CCD camera (C)

was used.

Hanle measurements were performed with a polarization control consisting of the

two polarizers P2 and P3 together with the liquid crystal devices (LCW, LCR). The

light polarization at the QD is determined by the fixed vertical polarizer P2 and the

liquid crystal waveplate LCW, that allows for a retardance between λ/4 and 3λ/4.

The reflected PLE laser photons from the QD experience the same retardance and

are therefore linearly polarized after the LCW. To distinguish between co- and cross

polarized QD photons a liquid crystal rotator in combination with a linear polarizer

P3 were used. To filter out the strong pump laser in PLE experiments a longpass

filter (LP) at 950nm is needed. The typical broad lasing background of a diode laser

is filtered out with a short pass (SP) at 940nm, such that it is not overwhelming the

QD emission at ∼ 960nm.

For a resonant measurement of the Overhauser shift an independent polarization

control of the pump laser La2 and the probe laser La1 is needed, therefore we placed

the LCW directly after the resonant laser (see Fig. A.1). In combination with the

fixed vertical polarizer P1, a complete polarization control of the resonant laser is

achieved. With an additional λ/4 plate, the pump laser is circularly polarized that

allows for polarizing nuclear spins in the QD. Besides the long pass filter, a grating

(G) is needed in front of the collection fiber to filter out the strong pump laser.

The creation of the laser pulses necessary for the resonant measurement of the Over-

hauser shift was performed either with an acousto-optical modulator (AOM) from

Crystal Technology, model no.3080-125 or with an electro-optical modulator (EOM)

based intensity modulator from Jenoptik AM940b. In both cases the laser pulses

were fiber coupled to separate the alignement from the measurement setup presented

in Fig. A.1.

A.1. Experimental Setup CSetup for PLE measurements

SP

P2

P3

LCW

P1

LCW

CR

PD1

LCR LP

SP

C SIL

S

La1 La2

CF

l/4

P4

L1

G

Sh

Figure A.1.: Measurement Setup. Schematics of the setup used for the experi-ments in chapter 4. The elements drawn with dotted lines are optional, a detaileddescription is given in the text. A list of the different components is shown inTable A.1.

D Appendix

Item Description Manufacturer

Sp Spectrometer Princeton Instruments/Acton Research,SpectraPro 2759, f= 0.750 m

Detector Princeton Instruments, Spec-10:100BR/LN.1340×100 pixel back-illuminated CCD array

C CCD camera Watec, WAT-120NFM Flip mirror Thorlabs, BB1-E03P1,P2,P3, P4

Polarizer Thorlabs LPNIR050

LCR Liquid crystal polarizationrotator

Meadowlark, LPR-200-0915

Sh Shutter Uniblitz, LS6ZM2LCW Liquid crystal wave plate Meadowlark, LRC-200-0915L1 Achromatic lens Thorlabs, AC050-008-B-ML, NA=0.68SIL Solid immersion lens A.W.I. Industries. hemispherical ball lens,

ZrO2, uncoated.S Sample Described in detail in chapter 2PP Piezoelectric positioners Attocube, ANPx50/LT and ANPz50/LT.

Controller ANC150/3CR Bath cryostat Oxford instruments, Spectromag with 12T

split-coil magnetG Grating Ibsen Photonics, FSTG-NIR1500-903nmLa1,La2

Tunable diode laser 1 (forPLE, dR, RF)

Topitca DL pro, coarse tuning range 915-990nm, mode-hop free tuning range 20GHz,Pmax = 40 mW

Tunable diode laser 2 Toptica DL proDiode Laser (for imagingand PL)

Melles Griot, 56IC5210, λ = 780 nm, Pmax =50 mW

CF Collection fiber (for RFand dR)

Thorlabs P3-830A-FC-5

SP Short pass filter LC-3RD/940SPLP Long pass filter LC-3RD/950LP

Table A.1.: Detailed list of the most important elements used for the setup sketchedin Fig. A.1

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Acknowledgements XI

Acknowledgements

First of all, I would like to thank Atac for giving me the opportunity to do my PhD

in his group. He let me accomplish many interesting and challenging experiments,

broadened my horizon with several theoretical simulations and let me explore the

world of quantum optics with a lot of freedom.

I am very thankful also to Prof. K. Ensslin for investing his valuable time in co-

examining this work.

Without knowing anything about spectroscopy techniques at the beginning of my

PhD, it would have been difficult to start my project without a good teacher in the

lab. Many thanks go to Martin who taught me how to perform proper experiments

in an optics lab. In addition I would like to thank him for many inspiring discussion

and the critical proof-reading of this manuscript.

A special thank goes to Weibo who shared his knowledge about coherent rotation of

single spins with me. Without his help the coherence time measurements on single

hole spins would not have been possible.

I would like to thank Mena for all the discussions we had, her support on simulating

nuclear spins, the proof-reading of this thesis and for being a good friend. My office

mates Charles and Francesco I would like to thank for always being open for a

little chat about physics and other subjects. Many thanks to the whole group for

their support and the after-work activities like trips to the mountains, BBQs, movie

nights and of course the delicious Christmas dinners. Especially Kathi I would like

to thank not only for the perfect administration but also for actively improving the

social life in our group.

Moreover I thank Sarah, Tina, Lars, Andreas, Andres, Susanne, Kathi, Theo and

others for their friendship. The many WOKA lunches, fondue events, hiking trips

and much more were always a good distraction from my work at ETH.

A special thank goes to Kurt who gave me an introduction to welding and helped

me realizing an additional construction for the optical table.

Finally, I would like to thank my parents Rita and Hans for the endless support and

their understanding for my more and more infrequent visits. My brother Matthias

I thank for his clean room expertise and for always believing in me. I especially

would like to thank Philip for his love, encouragement and for always supporting

my plans.

Curriculum Vitae XIII

Curriculum Vitae

Personal DetailsName Priska Barbara Studer

Date of Birth 28th of October, 1984

Place of Birth Wolhusen, Switzerland

e-mail [email protected]

Education2009 – 2014 Ph.D. in Quantum Optics

Quantum Photonics group, ETH Zurich, SwitzerlandAdvisor: Prof. Dr. Atac Imamoglu

2006– 2009 Highschool teaching diploma in PhysicsETH Zurich, Switzerland

Mar – Jun 2008 Research intern at Yale University, CT, USA

2003 – 2008 Diploma in PhysicsETH Zurich, Switzerland

1997 – 2003 Kantonsschule Willisau, Maturaoption “Physics and Applied Mathematics”

QualificationsLanguages German, English, French

Software Mathematica, LabView, Matlab, Ansoft HFSS, Origin

Additional Experience2009 – 2014 Teaching assistant in Physics, ETH Zurich, tutoring undergraduate stu-

dents

2012 – 2013 Member of the organization comittee IONS-13 SwitzerlandInternational optics conference for PhD and Master students with 100 par-ticipants from 32 countries. Responsible for student contacts, organizationand planning of travel grants

2005 – 2007 Teaching assistant at the Institute of Applied Mathematics, ETH Zurich,tutoring undergraduate students

March, 2014

List of Figures XV

List of Figures

2.1. Growth of self-assembled QDs. . . . . . . . . . . . . . . . . . . . . . . 7

2.2. Confinement in an InGaAs QD. . . . . . . . . . . . . . . . . . . . . . 8

2.3. Field effect structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4. Charging n-doped samples with a single hole. . . . . . . . . . . . . . 11

2.5. Charging diagram measured in PL. . . . . . . . . . . . . . . . . . . . 12

3.1. dT/T measurement for X−. . . . . . . . . . . . . . . . . . . . . . . . 23

3.2. Opitcal detection of nuclear spin polarization in X+ . . . . . . . . . . 25

3.3. Comparison between dR and ∆ EOS. . . . . . . . . . . . . . . . . . . 27

3.4. Nuclear spin dynamics measured with p-shell emission. . . . . . . . . 28

3.5. Measurement scheme for pump probe with dR. . . . . . . . . . . . . . 30

3.6. dR measurement with software based Lock-In amplifier. . . . . . . . . 31

3.7. Pump probe measurement with dR. . . . . . . . . . . . . . . . . . . . 32

3.8. Pump probe measurement with dR for different Twait. . . . . . . . . . 33

3.9. Pump probe measurements with RF. . . . . . . . . . . . . . . . . . . 35

4.1. Scheme illustrating Hanle measurements. . . . . . . . . . . . . . . . . 39

4.2. Anomalous Hanle effect. . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3. Hanle measurements for different nuclear spin polarizations. . . . . . 42

4.4. Emission at the s-shell. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5. Hanle measurements for different sample structures . . . . . . . . . . 44

4.6. RF measurements sample A. . . . . . . . . . . . . . . . . . . . . . . . 46

4.7. Gate modulation at different frequencies. . . . . . . . . . . . . . . . . 47

4.8. CW RF measurement sample B. . . . . . . . . . . . . . . . . . . . . . 49

4.9. Extracted positions of the resonances. . . . . . . . . . . . . . . . . . . 50

4.10. Transition between Overhauser shift and Zeeman splitting. . . . . . . 51

4.11. Transition between Overhauser shift and Zeeman splitting. . . . . . . 52

4.12. Polarization dependence at Bx=0.45T. . . . . . . . . . . . . . . . . . 53

4.13. Nuclear field changes in Bx. . . . . . . . . . . . . . . . . . . . . . . . 56

5.1. Optical line width of X+. . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2. X+ in Voigt geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3. Scheme for Rabi oscillations. . . . . . . . . . . . . . . . . . . . . . . . 63

XVI List of Figures

5.4. Rabi oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5. Ramsey scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6. Delay time control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.7. Ramsey fringes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.8. T ∗2 measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.9. Ramsey scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.10. Spin echo measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.1. Measurement Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . C

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